n  If  n  I )  1  lilf  iltJij  injfliMij  J  li 
. .  11  u }  \  n  itmmmnmimmi 
ummiiminhmimir  ' 
^  nmmmmmm 
irinijjinu 


11  ( uj  nmuijwmmmnii 


mmmmmm 


1{IL„,. 


ijBilWffipl 

MnijjiiiuifmuiMJwimiJ* 

'•%iiiii|iipiM" 


mmmm^Si 


mmnmmmmm 
'xmmvmnammx 


(imimimMmBiim 

mmmmmmumm. .. 
nimriifijjijrumiJHHilj 


mwumw.munmm..., 

immmmimhuimmim . 

ujirr»}Um|iiutH|tivmjiiji- 

— •■ —mmmnm 

mmumm  i 


q^o 


IN   MEMORIAM 
FLORIAN  CAJORI 


UNIVERSITY  ALGEBRA 


BY 

C.  A.  VAN  VELZER  and  CHAS.  S.  SLIGHTER, 

PROFESSORS   m   THE   UNIVERSITY   OF   WISCONSIN. 


MADISON,    WIS. 

TRACY,   GIBBS  &  COMPANY. 
1892. 


COPYRIGHT, 

C.  A.  VAN  VELZER,   CHAS.  S,   SLIGHTER. 
1892. 


Tracy,  Gibbs  &  Co.,  Printers  and  Stereotypers. 


rs 


5 

PREFACE.  ' 

The  present  volume  is  a  modified  form  of  a  treatise  on 
•algebra  which  appeared,  as  a  preliminary  edition,  about 
four  years  ago.  The  results  of  the  authors'  experience 
with  the  preliminary  edition,  together  with  the  suggestions 
of  several  friends,  have  been  used  in  preparing  the  pres- 
ent work. 

The  authors  have  aimed  to  produce  a  work  which 
should  not  be  too  difficult  for  the  average  student  in 
college  and  universit}^  classes,  and  which  should  be  ex- 
tensive enough  to  include  all  that  is  given  of  algebra 
even  in  the  largest  American  institutions,  and  to  be 
useful  as  a  work  of  reference.  Although  not  intended 
for  absolute  beginners  in  algebra,  it  was  thought  best  to 
begin  at  the  beginning  and  to  make  the  book  complete  in 
itself. 

In  preparing  this  work,  the  authors  have  freely  con- 
sulted other  texts.  Those  which  have  been  found  most 
useful,  are  the  following:  Serret's  Algebre  Superieure, 
Comberouse*s  Algebre  Supelrieure,  Laurent's  Traite  d' 
Algebre,  Chrystal's  Algebra,  C.  Smith's  Treatise  on 
Algebra,  Oliver,  Wait  and  Jones'  Algebra,  and  Klempt's 
Lehrbuch  zur  einfiihrung  in  die  Moderne  Algebra. 

Many  thanks  are  due  to  Professor  Plorian  Cajori  of 
Colorado  College,  who  has  not  only  offered  valuable 
suggestions,  but  has  written  all  the  historical  notes. 

C.  A.  VAN  VELZKR. 
CHAS.  S.  SI.ICHTKR. 

Madison,   Wis.,  January,  i8pj. 


CONTENTS. 

PAGE. 

Chapter  I.  Introduction, 1 

Negative  numbers  and  quantities,  -  -  -      9 

Evaluation  of  expressions,      -  -  -  -  12 

Chapter  II.  Addition, 15 

Union  of  similar  terms  or  addition  of  monomials,    -    18 
Addition  of  expressions,  -  -  -  _  19 

Chapter  III.  Subtraction, 22 

Subtraction  of  expressions,  -  -  -  -    23 

Insertion  and  removal  of  parentheses,        -  -  26 

Chapter  IV.  Multiplication,  -         -         -         -         -       28 

Product  of  monomials,       -  -  -  -  -    33 

Product  of  a  polynomial  and  a  monomial,  -  34 

Product  of  a  polynomial  by  a  polynomial,  -  -    37 

Special  products,  -  -  -  _  _  4Q 

Chapter  V.   Division, 48 

Division  of  a  monomial  by  a  monomial,  -  -  51 

Division  of  a  polynomial  by  a  monomial,  -  51 

Division  of  a  polynomial  by  a  polynomial,      -  -  53 

The  fundamental  laws  of  algebra,      -  -         -  59 

Historical  note, 61 

Chapter  VI.  Mathematical  Induction,    -         -         -       63 

Chaptor  VII.  Factors  and  Multiples,      -         -         -       67 

Expressions  of  the  form  rt-^  —  ^c^,             -  -           -    71 

Expressions  of  the  form  x'^+ax-\-d,  -           -          71 

Expressions  of  the  form  <^;r 2  _^^_;,;^^^     .  .           -    73 

Expressions  of  the  form  «3 —<^3^        -  -           _          75 

Expressions  of  the  form  a^-\-d^,              -  -           -    76 

Expressions  of  the  form  x'^-\-a^x^-\-a^,  -           -          78 

Expressions  of  the  form  %"  —  «",             -  -           -    79 

Expressions  of  the  form  :?c"  + ««,       -  -           -          83 

Miscellaneous  factors,         -           -           -  -           -    87        . 

H.  C.  F.  of  expressions  easily  factored,  -           -          89 

H.  C.  F.  of  expressions  not  easily  factored,    -  -    91 

Lowest  common  multiple,       -           -  -           -          97 

L.  C.  M.  of  expressions  not  easily  factored,    -  -    99 

Chapter  VIII.  Fractions, 102 

Addition  of  fractions,         -  -  -  -  104 

Subtraction  of  fractions,  -  -  -  -        105 

Multiplication  of  fractions,  -  -  -  U)7 

Miscellaneous  fractions,  ...  .        uj 


CONTENTS. 


Chapter  IX.  Powers  and  Roots, - 

Square  of  a  binomial,         -  -  -  - 

Cube  of  a  binomial,       -  -  -  -  - 

Roots  of  monomials,  -  -  -  - 

Square  root  of  polynomials,   -  -  -  - 

Cube  root  of  polynomials,  .  -  - 

Chapter  X.  Simple  Equations,        -         -         - 

Literal  equations,    -  -  -  -  - 

Symbolic  expression,     -  -  -  -  - 

Problems,       ------ 

General  equation  of  first  degree,       -  -  - 

Generalized  problems,       -  -  -  - 

Historical  note,     ------ 

Chapter  XI.  Simultaneous  Equations, 

Elimination  by  subtraction,    -  -  -  - 

Elimination  by  comparison,         -  -  - 

Elimination  by  addition  and  subtraction, 
Special  expedients,  -  _  -  - 

Simultaneous  equations  containing  three  unknown 
numbers,  ------ 

Literal  simultaneous  equations, 

Problems,  ------ 

General  system  with  two  unknown  numbers, 

Chapter  XII.   Quadratic  Equations, 

Pure  quadratic  equations,  -  -  - 

Solution  by  factoring,  -  -  -  -  - 

Affected  quadratics,  -  -  -  - 

Literal  quadratic  equations,  -  -  -  - 

Solution  by  factoring,        -  -  -  - 

Problems  leading  to  quadratic  equations, 
Equations  solved  like  quadratics. 


120 
121 
123 
128 
132 


141 
142 
145 
150 
151 
154 


156 
157 
159 
160 

163 
168 
169 
173 

177 
179 
180 
184 
185 
186 
192 

Chapter  XIII.  Theory  of  Quadratic  Equations  and 

Expressions,  - 

Discussion  of  the  roots,  -  -  -  -        199 

Historical  note,         -----  204 

Chapter  XIV.  Theory  of  Indices,  -        -        -        - 

Fractional  exponents,        -  -  -  -  207 

Negative  exponents,      -----        214 
Zero  equations,        -----  221 

Chapter  XV.  Surds, 

Reduction  of  surds,       -  -  .  -  226 

Operations  on  surds,  -  -  -  -  230 

Rationalization  of  expressions  containing  surds,  233 

Functions  of  surds,  _  -  -  -  238 

Square  root  of  binomial  quadratic  surds,  -  -  241 


119 


135 


155 


176 


194 


206 


224 


CONTENTS. 

Chapter  XVI.  Single  Equations,  -        -         -  -     245 

Rationalization  of  equations,  -  _  _  256 
Graphic  representation  of  expressions  and  equations 

of  the  first  degree,  -  -  .  -  259 
Graphic  representation  of  quadratic  expressions  and 

equations,            -----  265 

Chapter  XVII.  Systems  of  Equations,  -         -  -     268 

Linear  quadratic  system,  -          -          -          -  275 
Systems  of  two  quadratics,     -           -          -           -  277 
Special  expedients,             _           _           _           -  280 
Graphic  representation  of  systems  of  linear  equa- 
tions,           283 

Problems,       ------  286 

Chapter  XVIIL  Theory  of  Limits,         -         -  ^289 

Theorems  on  limits.  -         -          -           -          -  293 

Indeterminate  forms,    -          -          -          -          -  298 

Chapter  XIX.  Ratio,  Proportion  and  Variation,    -     304 

Properties  of  ratios,            -           -           .           -  305 

Incommensurable  numbers     -           -           -           -  308 

Compound  ratios,    -          -           -           -          -  311 

Proportion,  -  -  -  -  -  -314 

Variation, 320 

Chapter  XX.   Progressions, 325 

Arithmetical  progressions,           -           -           -  325 

Geometrical  progressions,       -           -           -           -  331 

Infinite  geometrical  progressions,          -          -  336 

Hnrmonical  progressions,       -           -           -           -  338 

Miscellaneous  exercises,     -           -           -           -  340 

Chapter  XXI.  Arrangements  and  Groups,     -  -     343 

Chapter  XXII.   Binomial  Theorem,        -         -  -     367 

Properties  of  the  expansion,   -           -           -           -  370 

Multinomial  theorem.  -          -           -           -          .  373 

Historical  note, 375 

Chapter  XXIII.  Theory  of  Probabilities,       -  -     378 

Simple  probability,             .           -           -           -  379 

Total  probability, 383 

Compound  probability,      -           -          -          -  384 

Mathematical  expectation,      -          -          -          -  388 

Successive  trials,      -----  391 

Miscellaneous  exercises.          -          -          -          -  394 

Chapter  XXIV.  Convergence  and  Divergence  of 

Series,    --------     396 

Chapter  XXV.  Undetermined  Coefficients,     -  -     424 


CONTENTS. 

Chapter  XXVI.  Summation  of  Series,  -        -  -     438 

Series  reducible  to  the  form  of  t^-^ — zt^-\-Uj^—tt^ 

^u.^-ti^-^.,. ,        -          -          -          -  438 

Summatiou  by  undetermined  coefficients,  -          -  440 

Method  of  differences,        -           -           -           -  442 

Recurring  series,            -           -           _           -           -  445 

Chapter  XXVII.   Binomial  Theorem  for  Fractional 

and  Negative  Exponents,       -         -         -  -     453 

Chapter  XXVIII.   Continued  Fractions,         -  -     459 

Chapter  XXIX.    Derivatives,         -         -         -  -     482 
Chapter  XXX.     Incommensurable  Exponents  and 

Logarithms, 505 

Laws  of  incommensurable  indices,        -           -  509 

Logarithms,         ------  511 

Properties  of  logarithms,   -          -           -           -  513 

Characteristic  and  mantissa,             -           -           _  517 

Tables  of  logarithms,         -           -           -           -  522 

Computation  by  logarithms,   -           -           -           -  536 

Exponential  and  logarithmic  series,      -           -  538 

Historical  note,     ------  544 

Chapter  XXXI.  Complex  Numbers,      -         -  -     547 

Modulus  and  amplitude,          -           -           -           -  559 

Chapter  XXXII.  The  Rational  Integral  Function,    566 

Properties  and  constitution  of  derivatives  oif{pc)  595 

Binomial  coefficients,  -----  599 

Chapter  XXXIII.  Special  Equations,    -        -  -     602 

Reciprocal  equations,  -           -           -           -           -  602 

Binomial  equations,           -           -           -           -  606 

Cubic  equations,             -----  615 

Biquadratic  equations,       -           -           -           -  620 

Historical  note,     ------  625 

Chapter  XXXIV.  Separation  of  Roots,          -  -     627 

Sturm's  theorem,           -----  641 

Theorems  of  Fourier  and  Budan,          -          -  649 

Chapter  XXXV.  Numerical  Equations,          -  -     659 
Chapter  XXXVI.   Decomposition  of  Rational 

Fractions,       -------     669 

Rapid  method, 681 

•Chapter  XXXVII.  Graphic  Representation  of 

Equations, -  -     686 

Equations  of  the  form  y=f{x),   -          -          -  686 

Graphs  of  equations  of  the  form /(x,jj/)=0,         -  691 

Graphs  of  quadratic  systems,       -           -           -  692 

Chapter  XXXVIII.   Determinants,         -         -  -     695 

Rationalization  of  any  algebraic  expession,      -  731 


I 


CHAPTER  I. 

INTRODUCTION. 

1.  Number  and  Quantity.  Anything  which  can  be 
measured  by  a  unit  of  the  same  kind  is  called  a  Quantity. 
Thus,  10  bushels  is  a  quantity,  the  unit  being  a  bushel, 
and  this  unit  taken  10  times  gives  the  quantity  10  bushels. 
Also,  10  cords  is  a  quantity,  the  unit  in  this  case  being 
one  cord,  and  this  unit  taken  10  times  gives  the  quantity 
10  cords.  Also,  the  abstract  number  10  is  a  quantity, 
the  unit  in  this  case  being  the  abstract  number  1,  and 
this  unit  taken  10  times  gives  the  quantity  10.  Of  course 
the  unit  itself  is  a  quantity. 

The  word  quantity  as  above  defined  plainly  includes 
number,  but  while  a  number  is  a  quantity,  a  quantity  is 
not  always  a  number.  Five  miles  would  always  b<^  called 
a  quantity  and  never  be  called  a  number,  but  the  number 
5  may  be  called  either  a  number  or  quantity  indifferently. 

The  word  quantity  is  usually  used  as  here  explained, 
but  some  writers  on  Algebra  never  use  the  word  quantity 
to  include  number. 

2.  Letters  used  for  Numbers.  In  Algebra  letters 
are  used  to  represent  or  stand  for  numbers.  Any  letter 
may  be  used  to  represent  any  number  provided  the  same 
letter  represents  the  same  number  throughout  the  same 
discussion. 

The  answer  to  a  problem  in  Algebra  is  often  something 
like  5  miles  or  4  tons  or  3  dollars  or  some  other  concrete 
quantity,  but  the  reasoning  is  always  conducted  by  nurn- 
bers,  and  so  the  letters  used  in  Algebra  always  represent 

1  — U.  A. 


2  UNIVERSITY   ALGEBRA. 

numbers,  and  the  result  reached  is  the  number  of  miles  or 
tons  or  dollars  or  whatever  it  may  be,  and  the  name  of 
the  thing  we  are  considering  may  be  added  at  the  end  to 
the  nufnber  we  have  obtained  by  solving  the  problem. 

3.  The  Sign  of  Addition  is  +,  read  ''plus.''  When 
this  sign  is  placed  between  two  numbers  it  signifies  that 
the  two  numbers  are  to  be  added  together.  Thus  a-\-b 
denotes  the  sum  of  the  numbers  represented  by  a  and  b. 
When  two  or  more  numbers  are  added  together  the  result 
i'§  called  the  Sum. 

4.  The  Sign  of  Subtraction  is  —,  read  ''minus.'' 
When  this  sign  is  placed  between  two  numbers  it  signi- 
fies that  the  second  of  the  two  numbers  is  to  be  subtracted 
from  the  first.  Thus  a—b  denotes  the  result  obtained  by 
subtracting  the  number  represented  b}^  b  from  the  number 
represented  by  a.  When  one  number  is  subtracted  from 
another  the  result  is  called  the  Difference  or  Remainder. 

5.  The  Sign  of  Multiplication  is  X  ,  read  "times"  or 
''into"  or  "multiplied  by"  When  this  sign  is  placed  be- 
tween two  numbers  it  signifies  that  the  two  numbers  are 
to  be  multiplied  together.  Thus  aY^b  denotes  the  result 
obtained  by  multiplying  together  the  numbers  represented 
by  a  and  b.  When  two  or  more  numbers  are  multiplied 
together  the  result  is  called  the  Product. 

6.  In  Algebra  it  is  usual  to  omit  the  sign  X  except 
between  numbers  represented  by  Arabic  numerals.  Thus, 
instead  of  writing  7  X  <^,  we  write  simply  lb  (read  '  'seven 
b"),  and  instead  of  ^xaxb  we  write  simply  Zab  (read 
"three  ab"^.  Sometimes  a  dot  is  used'  instead  of  the 
sign  X  to  indicate  multiplication.    Thus  2.3.7  means  the 


INTRODUCTION.  3 

same  as  2  X  3  X  7.  The  dot  should  not  be  used  when  two 
numbers  represented  by  Arabic  numerals  are  multiplied 
together  for  fear  of  confusing  it  with  a  decimal  point. 

7.  Factor.  When  two  or  more  numbers  are  multiplied 
together  to  form  a  product,  each  of  the  numbers  or  the 
product  of  any  number  of  them  is  called  a  Factor  of  the 
product.  Thus,  if  5,  a  and  b  are  multiplied  together, 
the  product  is  5^^,  and  the  factors  of  the  product  are 
5,  a,  b,ha,  ab^  5b, 

8.  Any  factor  of  a  product  is  called  the  Coefficient  or 
Co-factor  of  the  product  of  the  remaining  factors.  Thus, 
in  the  product  a^ be,  a"^  is  the  coefficient  or  co-factor  of  be, 
a^b  is  the  coefficient  of  c,  ab  is  the  coefficient  of  ac,  etc. 

9.  Numerical  and  Literal  Factors.  When  a  prod- 
uct is  made  up  partly  of  numbers  represented  by  figures 
and  partly  of  numbers  represented  by  letters,  we  call  the 
numbers  represented  by  figures  Numerical  Factors  and 
those  represented  by  letters  Literal  Factors. 

10.  Numerical  Coefficient.  The  product  of  all  the 
numerical  factors  of  any  product  is  of  course  the  coefficient 
of  the  product  of  all  the  literal  factors,  and  the  former  is 
often  called  the  Numerical  Coefficient  of  the  latter. 

11.  Power  of  a  Number.  When  a  product  consists  of 
the  same  number  repeated  any  number  of  times  as  a 
factor,  the  product  is  called  a  Power  of  that  number  and 
is  usually  written  in  a  simplified  form.     Thus  : 

aa  is  written  a'^ ,  read  ''a  square^'  or  ''  second  power  of  a*' ^ 
aaa  is  written  a^ ,  read  •*«  eube'^  or  ''third  power  of  a-y 
aaaa  is  written  a^,  read  ''a  fourth''  or  '  fourth  power  of  a  ;^^ 
and  so  on. 


4  UNIVERSITY   ALGEBRA. 

12.  Exponent  or  Index.  The  small  figure  written 
above  and  to  the  right  of  a  number  is  called  the  Exponent 
or  Index  of  the  power;  it  shows  how  many  times  the 
number  occurs  as  a  factor  in  the  power. 

According  to  this  notation,  a^  means  that  a  is  used  once  as  a 
factor,  but  when  a  number  is  used  only  once  as  a  factor  it  is  cus- 
tomary to  omit  the  exponent  and  write  simply  a  instead  of  ^i. 

13.  The  Sign  of  Division  is  -r-,  read  ''divided  by,'' 
When  this  sign  is  placed  between  two  numbers  it  signifies 
that  the  first  number  is  to  be  divided  by  the  second  num- 
ber. Thus,  a-T-b  denotes  the  result  obtained  by  dividing 
a  by  b. 

Usually,  however,  division  is  indicated  by  a  fraction 
with  the  dividend  above  the  line  and  the  divisor  below. 

Thus,  a-T-b  is  written  -r,  and  when  written  in  this  form  it 
o 

is  often  read  '  'a  over  b. ' ' 

14.  It  is  stated  in  Art.  2  that  letters  are  used  to  rep- 
resent numbers.  Several  letters  may  occur  in  the  same 
problem,  each  letter  standing  for  some  number.  Not  only 
may -several  letters  occur  in  the  same  problem,  but  several 
letters  may  be  combined  by  means  of  algebraic  signs  and 
this  combination  of  letters  also  stands  for  some  number. 
Thus,  if  ^  stands  for  6,  b  for  10,  and  c  for  5,  then  a-^bc-\-b'^ 
stands  for  6  +  50+100,  or  156. 

Anything,  whether  short  and  simple  or  long  and  com- 
plicated, which  stands  for  some  number  is  called  an 
Expression. 

The  number  which  an  expression  stands  for  is  called 
the  Value  of  the  expression. 

15.  When  an  expression  is  broken  in  parts  just  before 
each  of  the  signs  +  and  — ,  each  part  thus  formed  is 


INTRODUCTION.  5 

called  a  Term  of  the  expression.  Thus,  in  the  expres- 
sion Sad-i-4c'^—2e—ld  the  terms  are  Sad,  +4:C^,  —2^,  —15. 
We  speak  of  the  terms  of  an  expression  in  a  manner  somewhat 
analogous  to  the  way  in  which  we  speak  of  the  syllables  of  a  word. 
A  syllable  may  nol  convey  an  intelligible  idea  when  taken  by  itself, 
but  when  joined  to  other  syllables  to  make  up  a  word,  the  whole  word 
does  convey  a  definite  idea.  So  a  term  taken  by  itself  may  not,  at 
this  stage,  express  any  idea,  but  the  whole  expression  does  convey  an 
idea.  See  Art.  14.  Indeed,  each  terra  would,  even  now,  convey  a 
definite  idea  were  it  not  for  the  sign  +  or  —  which  always  goes  with 
each  term  after  the  first. 

16.  The  terms  of  an  expression  which  have  the  sign  + 
or  no  sign  at  all  are  called  Additive  Terms,  and  those 
terms  which  have  the  sign  —  are  called  Subtractive 
Terms.  Thus,  in  the  expression  n-\-2d—4:a-\-5a—7d, 
the  terms  n,  -}-2d,  and  -\-5a  are  additive  terms,  and  — 4« 
and  —73  are  subtractive  terms. 

In  comparing  several  additive  terms  they  are  spoken  of 
as  terms  of  the  same  sign,  and  several  subtractive  terms 
are  also  spoken  of  as  terms  of  the  same  sign ;  but  when 
additive  and  subtractive  terms  are  spoken  of  together, 
they  are  said  to  be  terms  of  imlike  or  opposite  signs. 

Notice  that  when  we  speak  of  the  signs,  without  any  further  quali- 
fication, it  is  only  the  first  two  of  the  four  fundamental  signs  of 
Algebra,  +,  — ,  X,  and  -h,  that  we  have  reference  to. 

17.  Terms  whose  literal  parts  are  identical  and  whose 
signs  and  numerical  coefficients  may  or  may  not  differ 
are  called  Similar  Terms.  Thus,  in  the  expression 
9a2<^— 3<aj^^— a^^-f-4a^2,  the  terms  ^a'^b  and  —a'^b  are 
similar;  also  the  terms  —Sab'^  and  +4^^^  are  similar;  but 
^a'^b  and  +Aab'^  are  not  similar. 

18.  An  expression  which  consists  of  but  one  term  is 
called  a  Monomial,  and  one  which  consists  of  more  than 


6  UNIVERSITY   ALGEBRA. 

one  term  is  called  a  Polynomial.  Thus,  6ad  is  a  mono- 
mial and  2^2__4^_|_2  is  a  polynomial. 

A  polynomial  which  consists  of  just  two  terms  is  called 
a  Binomial,  and  one  which  consists  of  just  three  terms  is 
called  a  Trinomial.  Thus,  Sa—ix^  is  a  binomial  and 
6x—4:xy-\-S  is  a  trinomial. 

While  binomials  and  trinomials  are  each  polynomials,  yet  it  is 
usual  to  apply  the  word  polynomial  only  to  expressions  of  more  than 
three  terms. 

19.  The  Degree  of  a  Monomial  wM  respect  to  any 
letter  or  letters  it  may  contain  is  the  sum  of  the  exponents 
of  the  letters  named;  unity  being  always  understood 
when  no  exponent  is  written.  Thus,  hab'^x^y^  is  of  the 
first  degree  with  respect  to  a,  of  the  second  degree  with 
respect  to  b,  of  the  third  degree  with  respect  to  x,  of  the 
fourth  degree  with  respect  to  y,  of  the  seventh  degree 
with  respect  to  x  and  j/,  of  the  seventh  degree  with  re- 
spect to  a,  b,  and  j/,  etc. 

1.  What  is  the  degree  of  ^a'^b^x^y^  with  respect  to  ^? 
What  with  respect  to  :i;?  What  with  respect  to  j/?  What 
with  respect  to  a,  b,  and  :r?  What  with  respect  to  x  and  j/? 

2.  What  is  the  degree  ot  ^a'^x'^y^  with  respect  to  x'> 
What  with  respect  to  jj/?  What  with  respect  to  x  and  j/? 
What  with  respect  to  a  and  x1  What  with  respect  to  a, 
X,  and  J^'? 

20.  When  the  degree  of  a  monomial  is  spoken  of  with- 
out specifying  the  letters  with  respect  to  which  the  degree 
is  taken,  it  is  usually  understood  to  mean  the  degree  with 
respect  to  all  the  letters  it  contains,  and  is  then  equal  to 
the  number  of  literal  prime  factors,  or  what  is  the  same 
thing,  the  sum  of  all  the  exponents  of  the  letters  in  the 
expression. 


INTRODUCTION.  7 

21.  The  Degree  of  a  Polynomial  with  respect  to  any 
letter  or  letters  it  may  cojitain  is  the  degree  of  that  one  of 
its  terms  whose  degree  with  respect  to  the  specified  letters 
is  highest.  Thus,  a'^x^ -\-abc^+e'^x^y^  is  of  the  second 
degree  with  respect  to  a,  because  the  first  term  is  of  the 
second  degree  with  respect  to  a  and  neither  of  the  other 
terms  is  of  so  high  degree  with  respect  to  a. 

The  same  expression  is  of  the  fourth  degree  with  re- 
spect to  X,  because  the  third  term  is  of  the  fourth  degree 
with  respect  to  x  and  neither  of  the  other  terms  is  of  so 
high  degree  with  respect  to  x. 

3.  What  is  the  degree  of  ax'^y-\-bxy'^-\-x'^y'^  with  re- 
spect to  ^?  What  with  respect  to  jk?  What  with  respect 
to  X  and  yl  What  with  respect  to  a?  What  with  respect 
to  a  and  b'> 

4.  What  is  the  degree  of  x^y+xy^+x^  with  respect 
to  jr?  What  with  respect  to  y?  What  with  respect  to 
X  and  y? 

Ans.  Five,  for  the  degree  with  respect  to  x  alone  is  5,  and  the 
term  that  determines  the  degree  has  no  y,  so  it  leaves  the  degree  5. 

5.  What  is  the  degree  of  x^  +  ax^y+ dxy'^-j- a  by  ^  with 
respect  to  x?  What  with  respect  toy?  What  with  respect 
to  X  and  y? 

6.  What  is  the  degree  of  a'^bx+d^xy^+cxy^  with  re- 
spect to  a?  What  with  respect  to  jf  ?  What  with  respect 
toy}  What  with  respect  to  a  and  x?  What  with  respect 
to  a,  b,  c,  X,  and  jj/? 

22.  When  the  degree  of  a  polynomial  is  spoken  of  with- 
out specifying  the  letters  with  respect  to  which  the  degree 
is  taken,  it  is  usually  understood  to  mean  the  degree  with 
respect  to  all  the  letters  it  contains,  and  is  then  equal  to 


8  UNIVERSITY   ALGEBRA. 

the  number  of  literal  prime  factors  in  that  term  which 
contains  the  greatest  number  of  such  prime  factors. 

23.  The  Sign  of  Equality  is  =,  read  ''equals''  or 
**w  equal  to.'' 

24.  The  statement  of  equality  which  exists  between 
two  expressions  is  called  an  Equation,  and  the  parts  on 
either  side  of  the  sign  =  are  called  the  Members  of  the 
equation.  The  expression  on  the  left-hand  side  of  the 
sign  =  is  called  the  Left  or  First  Member,  and  the  ex- 
pression on  the  right-hand  side  of  the  sign  =  is  called 
the  Right  or  Second  Member. 

25.  The  Sign  of  Inequality  is  >,  read  ''greater  than' ^ 
or  <,  read  ''less  than."  Thus  a^b  signifies  that  a  is 
greater  than  b,  and  a<ib  signifies  that  a  is  less  than  b. 

Whenever  this  symbol  is  used  the  point  of  the  angle  is 
toward  the  lesser,  and  the  opening  of  the  angle  toward 
the  greater  ot  the  two  numbers. 

26.  The  Sign  of  Aggregation.  When  an  expression 
consisting  of  two  or  more  terms  is  to  be  looked  upon  as  a 
whole  it  is  enclosed  in  a  parenthesis  or  in  brackets.  Thus, 
25— (3-f  2)  means  that  the  sum  of  3  and  2  is  to  be  sub- 
tracted from  25.  The  use  of  the  parenthesis  in  Algebra 
is  quite  common,  and  it  always  serves  to  show  that  the 
expression  which  it  encloses,  whether  simple  or  compli- 
cated, is  to  be  looked  upon  as  a  single  number  just  as 
though  it  were  represented  by  a  single  symbol. 

27.  When  several  parentheses  are  used  one  within 
another,  they  are  often  made  of  different  shapes  and 
sometimes"  of  different  sizes  to  prevent  confusion.     Some 


INTRODUCTION.  9 

of  the  forms  used  are,  (  ),  [  ],  {  } .  Sometimes  a  horizontal 
line,  called  a  Vinculum  or  bar,  is  drawn  above  an  ex- 
pression instead  of  using  a  parenthesis.  Thus,  a-^x—y 
means  the  same  as  a-{-(x—y). 

28.  The  Sign  of  Continuation  is  .  .  .  ,  read  ''and  so 
ony  Thus,  1-f  i+i+-§-4-.  ...  is  read  ''1  plus  \  plus  \ 
plus  \  plus  and  so  on,''  This  means  that  other  terms  are 
to  be  written  which  follow  one  another  according  to  the 
law  made  obvious  by  the  terms  already  written,  viz. :  in 
this  case,  each  term  is  one-half  the  term  before  it. 

29.  The  Sign  of  Deduction  is  .*.,  read  ''hence"  or 
"  therefore  y 

NEGATIVE   NUMBERS   AND   QUANTITIES. 

30.  In  Algebra  we  are  often  called  upon  to  distinguish 
between  quantities  which  are  directly  opposite  each  other; 
as,  for  instance,  degrees  above  zero  from  degrees  below 
zero  on  a  thermometer  scale,  distance  north  from  distance 
south,  gain  from  loss,  etc. 

31.  Thus  it  is  seen  that  we  often  have  to  consider  some- 
thing besides  magnitude  or  amount,  and  this  something 
we  may,  for  want  of  a  better  word,  call  Direction.  The 
quantities  spoken  of  in  the  last  article  as  being  opposite 
each  other  are  simply  opposite  in  direction.  For  example, 
degrees  above  and  degrees  below  are  in  opposite  direc- 
tions, distance  north  and  distance  south  are  in  opposite 
directions,  gain  and  loss  are  in  opposite  directions,  time 
before  and  time  after  are  in  opposite  directions,  etc. 

32.  Opposite  directions  are  distinguished  by  means  of 
the  signs  -f-  and  — ,  ^.  ^.,  if  +10°  means  10°  aJ)ove  zero. 


lO  UNIVERSITY   ALGEBRA. 

then  —10°  means  10°  below  zero,  and  if  +10  miles  means 
10  miles  east,  then  — 10  miles  means  10  miles  west,  and  if 
+  10  dollars  means  10  dollars  gain,  then  —10  dollars 
means  10  dollars  loss,  and  if  + 10  represent  10  units  of  any 
kind  in  ^zM<?r  direction,  then  —10  represents  10  units  of 
the  same  kind  in  the  opposite  direction. 

33.  We  see  from  this  that  the  signs  Plus  and  Minus 
have  a  double  use  in  Algebra;  first,  to  indicate  the 
operations  of  addition  and  subtraction  respectively;  second, 
to  distinguish  between  opposite  directions  as  just  de- 
scribed. There  is  no  danger  of  confusion  arising  from 
this  double  use;  the  context  will  always  make  it  clear  in 
which  sense  the  signs  are  used. 

34.  If,  in  any  concrete  quantity,  we  abstract  the  con- 
crete unit,  we  have  left  a  7itcmbe7-,  and  hence  the  numbers 
we  have  to  deal  with  in  Algebra  are 

^-5-4-3-2-1     0+1  +  2  +  3  +  4  +  5 

and  intermediate  numbers. 

The  row  of  numbers  here  give  is  sometimes  called  the 
Algebraic  Scale.  It  extends  indefinitely  in  both  direc- 
tions from  0. 

35.  Numbers  to  the  right  of  0  in  the  above  scale  (Art. 
34)  are  called  Positive  Numbers,  and  those  to  the  left  of 
0  are  called  Negative  Numbers;  that  is,  Arabic  numerals 
preceded  by  a  +  sign  are  positive  numbers,  and  Arabic 
numerals  preceded  by  a  —  sign  are  negative  numbers. 

A  number  with  no  sign  at  all  before  it  is  considered  the 
same  as  though  the  +  sign  were  written  before  it,  e.  g. , 
6  is  considered  the  same  as  +6  and  a  the  same  as  +a. 

36.  In  Algebra  numbers  are  represented  by  letters,  but 
a  letter  is  just  as  apt  to  represent  a  number  to  the  left  of 


INTRODUCTION.  1 1 

0  in  the  above  scale  (Art.  34)  as  it  is  to  represent  one  to 
the  right  of  0;  so  that,  while  in  the  case  of  a  number 
represented  by  figures  we  can  tell  whether  the  number 
is  positive  or  negative  by  the  sign  preceding  it,  yet,  in  the 
case  of  a  number  represented  by  letters,  we  cannot  tell  by 
the  sign  before  it  whether  the  number  is  positive  or 
negative. 

If  we  speak  of  the  number  5  we  know  that  it  is  positive, 
but  if  we  speak  of  the  number  a  we  do  not  know  by  the 
sign  before  it  whether  it  is  positive  or  negative. 

We  know  that  —5  is  negative,  but  we  do  7iot  know 
that  —  <2  is  negative. 

A  letter  with  a  fniniis  sign  before  it  always  represents  a 
number  of  the  opposite  kind  from  that  represented  by  the 
same  letter  with  a  plus  sign  or  no  sign  at  all  before  it. 
Thus,  if  a=3,  then  — a=  —3,  and  if  «=  —3,  then  —^=3. 

37.  The  student  should  notice  the  distinction  between 
additive  and  subtractive  terms  on  the  one  hand,  and 
positive  and  negative  numbers  on  the  other.  — 5  is 
always  a  subtractive  term,  and  it  is  also  always  a  negative 
number;  —  ^  is  always  a  subtractive  term,  but  may  be 
either  a  positive  or  negative  number,  depending  upon  the 
value  of  a.     See  Art.  16. 

38.  If  to  a  positive  number  we  attach  a  concrete  unit 
we  obtain  a  positive  quantity,  and  if  to  a  negative  number 
we  attach  a  concrete  unit  we  obtain  a  negative  quantity. 

39.  The  value  of  a  number  independent  of  the  sign  is 
sometimes  called  the  Absolute  Value  of  the  number. 
Thus,  +3  and  —3  have  the  same  absolute  value,  viz.:  3. 

40.  Looking  at  the  above  scale  (Art.  34)  it  is  evident 
that  of  any  two  positive  numbers  the  one  at  the  right  is ' 


12  UNIVERSITY    ALGEBRA. 

greater  than  the  other,  or  the  one  at  the  left  is  less  than 
the  other,  e.  g.,  10>6  or  6<10. 

Now  it  is  found  convenient  to  extend  the  meaning  of 
the  words  ' '  less  than  ' '  and  ' '  greater  than  ' '  so  that 
throughout  the  whole  scale  any  number  will  be  greater  than 
any  number  to  the  left  of  it,  and  less  than  any  number  to 
the  right  of  it. 

Thus  we  would  say  that 

~5<-3  and  -2<0. 
It  should  be  carefully  noticed  that  this  is  a  technical  use 
of  the  words  ' '  greater  than  ' '  and  ' '  less  than  ' '  and  con- 
forms to  the  popular  use  of  these  words  only  when  the 
numbers  are  positive. 

Of  course  it  would  be  wrong  to  say  that  —2  is  less  than 
0  if  we  use  '*  less  than  "  in  the  popular  sense,  because  no 
number  can  be  less  than  nothing  at  all,  in  the  popular 
sense  of  "less  than."  But  if  we  keep  in  mind  that  the 
words  are  used  in  a  technical  sense,  there  is  no  objection 
to  such  inequality  as  —  2<0. 

KVAI^UATION  OF  KXPRKSSIONS. 

41.  When  the  values  of  all  the  letters  of  an  expression 
are  given  the  value  of  the  expression  may  be  found  by 
putting  in  place  of  the  letters  the  given  values  and  per- 
forming the  indicated  operations. 

For  example,  if  ^=5,  <^=4,  and  ;z^lO,  then  the  value 
oi  (a^ —Zb'^^n-\- an'^  is  easily  found  as  follows  : 

a3=53  =  125.     ^2=42_i6^  ^nd  S^^^g^  16=48. 
...  a3-3^2=,i25_48=77, 
.-.  (^3 _3^2)^_77x  10=770. 
Also,  ;z2  =  i02  =  l00, 

.-.  a7^2  =  5x  100=500, 
...  («3_3^2)^+^^2_770-f-500=1270. 


INTRODUCTION.  1 3 

KXAMPI.KS. 

Find  the  value  of  each  of  the  following  five  expressions 
for  the  values  of  a  and  b  given : 

1.  [- |-+2/^)-2a^whena=5,  ^=3. 

\  a-\-o  / 

~aAl) 

2.  — ; when  a=3,  <5=2. 

a-\-b 

3.  (3a2-4^2)(^3^2_|.4^2>)  ^j^en  a=3,  ^=2. 

5-  ftl|J+^^'-^')  w^^^  ^=2'  ^=^- 
Find  the  value  of  each  of  the  following : 

6.  a^— 3a -  +  2^  —  14  when  <2=4. 

7.  {a''-—^\a^-^)^\i^na=^. 

8.  ^3  — 2^2-^  +  2  when  ^=5. 

g.   (a--2)(a  — 1)(<2  +  1)  when  a=5. 

^3_27     ^3+27     ^ 

10. ^ — y-  when  <2=4. 

a — o         a-\-€> 

11.    ^i — ;r  when  a=o. 

a — 2       a-f-2 

12.  <2'^ — ^2 — 5^ — 50  when  <3^=6. 

13.    —-\ — r  when  a=l(J. 

a—1       a-j-1 

14.  If  10:i:=50,  what  is  the  value  of  jt? 

15.  If  10x—2o,  what  is  the  value  of  ^? 

16.  If  o.r=50,  what  is  the  value  of .;»:? 

17.  If  7-r=42,  what  is  the  value  of  .r? 

18.  If  7:r+2=44,  what  is  the  value  of.;*:? 

19.  If  5:r— 10=40,  what  is  the  value  of  .r? 

20.  If  5.^+5=55,  what  is  the  value  of  ;r? 


14  UNIVERSITY    ALGEBRA. 

42.  A  careful  inspection  of  examples  6  to  20  above 
shows  that,  when  there  is  only  one  letter  in  an  expression, 
two  cases  may  arise  :  first,  the  value  of  the  letter  may  be 
given  and  the  value  of  the  expression  required ;  second, 
the  value  of  the  expression  may  be  given  and  the  value 
of  the  letter  required. 

In  the  first  case  the  value  of  the  letter  is  kfiown  or 
given,  and  in  the  second  case  the  value  of  the  letter  is 
unknown  or  required. 

Thus  we  see  that  in  Algebra  there  are  two  kinds  of 
numbers,  called  respectively  Known  and  Unknown, 
either  of  which  may  be  represented  by  a  letter ;  and,  as  it 
is  possible  for  both  kinds  of  numbers  to  appear  in  the 
same  discussion,  it  is  customary  to  distinguish  between 
them  by  representing  the  known  numbers  by  the  first 
and  intermediate  letters  of  the  alphabet,  and  the  unknown 
numbers  by  the  last  letters  of  the  alphabet. 

To  determine  the  values  of  letters  when  the  values  of 
the  expressions  which  contain  these  letters  are  given, 
is  one  of  the  most  important  questions  of  Algebra.  Some- 
times this  can  be  done  easily,  sometimes  it  is  quite  diffi- 
cult, and  sometimes  it  cannot  be  done  at  all. 


CHAPTER  II. 

ADDITION. 

43.  Addition  is  the  process  of  finding  the  result  of 
taking  two  or  more  numbers  together.  The  result  is 
called  the  Sum,  and  the  numbers  added  are  called  the 
Summands. 

This  defines  addition  in  Arithmetic,  where  numbers  have  no  direc- 
tion, and,  as  we  shall  see,  is  sufficiently  broad  to  include  the  case  of 
negative  numbers. 

44.  To  indicate  that  the  sum  of  several  numbers  is  to 
be  found  we  supply  the  sign  +  to  numbers  having  no 
sign  and  then  write  the  numbers  down  one  after  another 
with  their  signs  unchanged.  Thus,  to  indicate  the  sum 
of  8,  —2,  — 5,  and  6,  we  write 

+  8-2-5  +  6. 

45.  When  we  add  +3  and  +8  we  get,  by  the  defini- 
tion, of  addition,  +3  +  8=+ll, 

because  the  result  of  taking  together  3  in  a  certain  direc- 
tion and  8  in  the  same  direction  is  11  in  that  direction. 

When  we  add  —3  and  +8  we  get,  by  the  definition  of 
addition,  —3  +  8= +5, 

because  the  result  of  taking  together  3  in  a  certain  direc- 
tion and  8  in  the  opposite  direction  is  5  in  the  latter 
direction. 

When  we  add  +3  and  —8  we  get,  by  the  definition  of 
addition,  +3—8=— 5, 

because  the  result  of  taking  together  3  in  a  certain  direc- 
tion and  8  in  the  opposite  direction  is  5  in  the  latter 
direction. 


1 6  UNIVERSITY   ALGEBRA. 

When  we  add  —3  and  —8  we  get 

because  the  result  of  taking  together  3  in  a  certain  direc- 
tion and  8  in  the  same  direction  is  11  in  that  direction. 

In  exactly  the  same  way  we  would  get,  by  the  defini- 
tion, the  sum  of 

-i-2a  and  +7<2,  or  2a  and  7a  in  same  direction,  =-}-9a 
—'2a  and  +7<2,  or  2a  and  7a  in  opposite  directions,  =  + 5a 
+  2«  and  —7a,  or  2a  and  7a  in  opposite  directions,  =  —5<2 
—  2a  and  —7a,  or  2a  and  7a  in  same  direction,         = — da 

In  general,  if  a  and  d  stand  for  any  two  numbers,  we 
have  as  above,  by  the  definition  of  addition, 
sum  of  -i-a  and  -\-d=-{-a-i-d 
sum  of  -\-a  and  —d=-\~a—d 
sum  of  — a  and  -j-d=—a-\-d 
sum  of  —a  and  —d=—a—d 

46.  Thus  it  follows  from  the  definition  of  addition  that 
the  sum  of  two  numbers  of  the  same  sign  is  the  arith- 
metical sum  of  their  absolute  values  with  the  common 
sign  of  the  summands,  and  the  sum  of  two  numbers  of 
opposite  signs  is  the  arithmetical  difference  of  their  abso- 
lute values  with  the  sign  of  the  summand  having  the 
greater  absolute  value. 

It  is  well  to  notice  one  important  difference  between  addition  in 
Arithmetic  and  addition  in  Algebra.  In  Arithmetic  addition  implies 
augmentation,  but  in  Algebra  this  is  not  necessarily  the  case. 

47.  To  add  6,  —7,  2,  and  —4  we  write 

+  6-7  +  2-4. 
To  find  a  shorter  form  for  this  sum,  we  may  put  -f  6  and 
—7  together  (as  in  Art.  45)  giving  —1 ;   then  we  may 
combine  this  result,  —1,  with  -f  2,  giving  +1 ;  then  this 


ADDITION.  17 

result,  +1,  with  —4,  giving  —-3  as  the  sum  of  6,  —7, 
2,  and  —4. 

Also,  to  add  ^,  —  ^,  — r,  and  ^  we  have 
-j-a—d—c+d. 
This  result  cannot  be  shortened. 

48.  Since  addition  is  the  process  of  finding  the  result 
of  taking  several  numbers  together,  it  follows  that  addi- 
tion may  be  performed  in  many  different  ways  by  taking 
the  numbers  zn  different  orders.  That  is,  addition  may  be 
performed  in  any  order.  This  is  called  the  Commutative 
Law  of  Addition.     It  may  be  symbolized  as  follows : 

—a^h^c—d=h—d-^c—a—c-{-h—d—ai  etc. 

49.  It  is  also  apparent  from  the  definition  that  addi- 
tion may  be  performed  in  many  different  ways  by  first 
adding  the  numbers  into  certain  groups  and  then  adding 
these  groups  together  to  get  the  required  sum.  Thus, 
the  sum  of  3,  5,  7  and  —3  may  be  found  by  first  adding 
3  and  5,  giving  8,  then  adding  7  and  —3,  giving  4, 
finally  adding  8  and  4  and  obtaining  12  as  the  sum  of  the 
original  numbers.     Symbolicly  this  may  be  written 

3  +  5  +  7-3=(3  +  5)  +  (7-3). 
Since  we  might  have  used  any  numbers,  as  a,  3,  — ^, 
—d,  and  e,  and  have  grouped  them  in  many  different 
ways,  we  may  write  more  generally 

a^rb—c—d\-c-^(a-\-h—c)-\-(^—d-\-e^ 
=  («  +  3)  +  (-^~^+^) 
=  (a-f  ^)  +  (— ^-— ^  +  <?,  etc.,  etc. 
This  fact,  that  addition  may  be  performed  by  grouping 
the   summands   in  many   different  ways,   is   called   the 
Associative  Law  of  Addition. 

2— U.  A. 


1 8  UNIVERSITY   ALGEBRA. 

50.  Removal  of  a  Parenthesis  preceded  by  the 
sign  +.  What  we  have  called  the  associative  law  of 
addition,  is  also  known  as  the  principle  of  the  insertion 
or  removal  of  a  parenthesis  preceded  by  the  plus  sign,  and 
may  be  stated  in  words  as  follows:  In  any  expression 
a  parenthesis  preceded  by  the  plus  sign  may  be  inserted^  or 
removed^  without  changing  the  value  of  the  expression, 

UNION  OF  SIMIIvAR  TKRMS   OR  ADDITION  OF  MONOMIAI^. 

51.  If  we  have  an  expression  containing  similar  terms ^ 
such  as 

2^32  j^c—Zab'^+hab'^  —  Vlab'^ 

we  may  change  the  order  of  the  terms,  (Art  48)  so  as  to 
bring  similar  additive  terms  in  one  group,  and  similar 
subtractive  terms  in  another  group,  thus  : 

c-\-  {2ab''  +  5a^2)  _(_  (_3^^2  _  I2ab'^), 
The  similar  additive  terms  may  now  be  replaced  by  a 
single  additive  term,  and  the  similar  subtractive  terms  by 
a  single  subtractive  term  (Art.  46)  giving 

•      c-^lab'^-lbabK 
The  similar  terms  may  now  be  replaced  by  a  single  term, 
giving  c—^ab'^, 

the  shortest  form  possible  for  the  given  expression. 

KXAMPI^KS. 

Shorten  the  following  expressions  as  much  as  possible 
by  a  careful  grouping  and  uniting  of  the  similar  terms: 

1.  2xy—^2+10xy—d>2+1^2-'12xy. 

2.  ^m—10ts—^n—Ats—2n  +  2m—^ts—^m. 

3.  ba-\'W^c—1d—2a—'db'^c+2d—2a+2b^c+M. 

4.  —10m+ll—bx—12—4:m—^x-\-l  +  ^x—bm, 

5.  9^-7^+3^— 8^+7^— 3^— 5^— 8^. 


ADDITION.  19 

7.  llxy-^-'lab—^^xy—^bab+ab+lO. 

8.  25— 25;i:+25j/+13— 30j/+20;i;-8. 

9.  7^2— 24-^2+2— 46r2+9>^2^5  +  ^^ 

10.  lSx—6y  +  8^—5x-\-9j/—llz—Sx—6j/-h^. 

11.  4m—62n  +  18x-^62m—Gx  +  4:2?z  +  10m-^lSn—14tx. 

12.  10m  +  ll—5a'^  —  12  +  4m-6a'^-{-l  +  18a'-  —  10m. 

13.  ^— 2^^+18r^— 14<^^-21^^— 3<^^+5^^. 

ADDITION   OF   KXPRKSSIONS. 

52.  When  two  or  more  expressions  are  to  be  added, 
we  may  enclose  each  expression  in  a  parenthesis,  supply 
the  sign  +  to  each  parenthesis,  and  then  write  the  paren- 
theses one  after  another. 

lyCt  us  find  the  sum  of  the  three  expressions 

x-\-y,  X — ^  and  2x-\-^y — 2. 
First,  we  enclose  these  in  parentheses,  write  them  one 
after  another  separated  by  plus  signs,  and  get 

(-^4-jr)  +  {x-2)  +  (2x-^^y-2). 
Second,  we  remove  parentheses  and  get 

x-\-y-\-x — ^4-2jr4-3j/— 2.  • 

Third,  we  arrange  these  terms  so  that  similar  terms  shall 
come  together,  and  get 

x+x+2x+y+Sy—^—2. 
Fourth,  we  unite  each  group  of  similar  terms  into  a  single 
term  and  get  4xi-4y—z—2. 

and  this  is  the  simplest  form  possible  for  the  sum  of  the 
three  given  expressions. 

53.  It  is  easy  to  see  that  we  can  find  the  sum  of  any 

number  of  expressions,  whatever  those  expressions  may 
be,  in  a  manner  similar  to  that  just  pursued.     To  do  this 


20  UNIVERSITY    ALGEBRA. 

we  enclose  each  expression  in  a  parenthesis  and  write  these 
parentheses  one  after  another  separated  by  plus  signs; 
then  we  remove  parentheses  from  the  expression.  Next 
we  arrange  the  terms  of  the  expression  thus  found,  so 
that  similar  terms  shall  come  together,  and  then  unite 
each  group  of  similar  terms  into  a  single  term. 

Kxampi,e;s. 
Find  the  sum  of  the  following  expressions: 

1.  ^m—4cst—r'^,     —llst+lm+^r'^  and  lOst—lr'^-'m. 

2.  3;t:2+^j/+3j/2,     x'^—Zxy-^y'^  and  3jr2  +  3jj/2. 

3.  Sxy+2y,     bxy—x,     Sx—5y  and  7xy—x—2y, 

4.  Sa—Ad—6cd+2e,  10<^+3^— 10^^  and  9^—20^+14^^. 

5.  6^+7— 4a— 5^,  6/5+7^— 4— 5a.  and  6a-\-7d—14:C—5. 

54.  Arrangement  of  \A^ork  in  Addition.  It  evidently 
comes  to  the  same  thing  if,  instead  of  writing  the  expres- 
sions one  after  another  within  parentheses,  as  in  Art.  53, 
we  write  them  one  below  another  without  parentheses, 
arranging  the  similar  terms  in  the  same  vertical  column. 
Then,  when  this  is  done,  we  may  draw  a  line  under  the 
last  expression,  and  the  example  is  arranged  in  exactly 
the  saine  form  as  in  Arithmetic.  Now  the  similar  terms 
in  each  column  may  be  combined  into  a  single  term,  and 
this  term  placed  under  the  line  as  one  term  of  the  sum. 
When  every  column  has  been  thus  treated  all  the  terms  of 
the  sum  are  found.  Thus,  suppose  we  are  required  to 
find  the  sum  of  ^ 

2^2+ 3^2  _5^^^     6^2-2^2  and  W^ -^a'^ -^cd. 

By  the  present  method  we  write 
2^2+3^2^5^^ 

6a2-2^2 

-4a2  +  8^2_4^^ 

4^2 +9^2  _9^^ 


ADDITION.  21 

Now  it  is  very  evident  that  we  have  here  exactly  the 
same  terms  to  combine  that  we  had  by  the  other  method, 
after  the  parentheses  had  been  removed,  and  the  similar 
terms  brought  together.  It  is  plain  that  the  only  differ- 
ence is  in  the  arrangement  of  the  work. 

l^XAMPIvKS. 
I.  2. 

4^2+5^3  +  932  2ad-  bc+bca 


3.  Add  2b+Sc—ba,     8^-33+4c,  and  Tb-lbc-la. 

4.  Add  n+2r-\-Zs—4:t,  r—As—bt—^n,  s—^t—^n^&r. 

5.  Add  x+Za+2b-c,  2y—Sb+2c+a,  2>z-Zc—2a—b. 

a-Sd-{-2c      +2r 

6.  Add  13:i;8— 4;»;2  — 6;ir  +  17,     22.^^+ 20.^2+ 3:r-- 10, 
2^2_i7^8_2;»;-14,  and  3^3_i2jtr2  + 12  +  5^1;. 

In  arranging  expressions  which  involve   different   powers  of  the 
same  number  it  is  usual  to  place   all   of   the   terms  containing  the 
highest   power  of  the  letter  in  the  first  column,   all  the  terms  con- 
taining the  next  highest  power  in  the  next  column,  and  so  on     Thus, 
this  example  would  generally  be  arranged  thus : 
13x3-  4x^-Qx+17 
22x3+20x;2+3x-10 
-17^3+  2x^-2x-U 
3x3-123c8+5jc+12 

7.  Add  24xy+15de'-12/g',     \Zfg--Z2xy,     V^xy-Zde, 
Me'-hxy'-2fg,  and  ^fg—'lxy. 

8.  Add  9|a»-7a2^+54^^2 +  111^3^   _7^2_^5^^+9^2^ 
and  7ia3-2^a2^_4^^2_i2^8-5a2+4a^-20/^2^ 


CHAPTER  III. 

SUBTRACTION. 

55.  Subtraction  is  the  process  of  undoing-  Sidditioii; 
that  is,  subtraction  is  the  process  of  finding  from  two 
given  numbers,  called  the  Minuend  and  Subtrahend, 
a  third  number,  called  the  Remainder  or  Difference, 
such  that  the  sum  of  the  subtrahend  and  remainder  shall 
equal  the  minuend. 

56.  Any  operation,  like  subtraction,  which  is  the  un- 
doing of  another  operation  is  often  called  the  Inverse  of 
the  other  operation. 

57.  Notation.  To  denote  that  one  expression  is  to  be 
subtracted  from  another  expression  we  enclose  each  in  a 
parenthesis  and  write  the  subtrahend  after  the  minuend 
with  a  minus  sign  between  them.  Thus,  to  indicate  that 
2:tr— 5  is  to  be  subtracted  from  x'^—4:X+2  we  write 

58.  Since  subtraction  is  defined  as  the  undoing  of 
addition,  it  follows  that  the  principles  of  subtraction 
must  be  based  on  those  of  addition. 

Thus:  +12-(+5)=+  7 

because  +  7 +  (+5)  =+12, 

and  +12-(-5)=  +  17 

because  + 17  +  (—5)  =  + 12. 

Also,  in  general,  a—[-\-h)-=a-h                           [1] 

because  a—b-\-(^-\-b)  =  a, 

and  a-{-h)--^a^h                            [2] 

because  a  +  b-{-{^—b)=a. 


SUBTRACTION.  23 

Since  a  and  b  may  stand  for  any  two  numbers,  we  may 
deduce  the  following  principle  from  [1]  and  [2] : 

To  subtract  a  given  number  from  any  expression^  we 
annex  the  nuTnber  with  its  sign  changed  to  the  expression 
from  which  it  is  to  be  subtracted, 

59.  Associative  La^v  of  Subtraction.  The  associa- 
tive law  of  addition  states  that  we  may  add  a  given  ex- 
pression to  another  by  adding  the  expression  as  a  whole 
or  by  adding  its  terms  in  succession.  Hence,  since  sub- 
traction is  the  inverse  of  addition,  to  subtract  a  given 
expression  from  another  we  have  merely  to  subtract  its 
separate  terms  in  succession.     Thus: 

=-a — b-\-c — d, 

60.  Removal  of  a  Parenthesis  preceded  by  the 
sign  — .     Comparing  the  two  sides  of  the  equality 

a—{b—c-\-d')^=a—b-\-c—d, 
we  derive  the  principle  of  the  insertion  or  removal  of  a 
parenthesis  preceded  by  the  minus  sign : 

A  parenthesis  preceded  by  the  minus  sign  may  be  inserted 
or  removed,  provided  that  the  sign  of  every  term,  within  the 
parenthesis  be  changed. 

SUBTRACTION   OF  KXPRKSSIONS. 

61.  I^et  us  find  the  difference  between 

Sx^—4:xy+5y^  and  2x^—'2xy-'4y^. 
First,  we  enclose  each  of  these  expressions  in  a  paren- 
thesis, and  write  the  subtrahend  after  the  minuend  with 
a  minus  sign  between  them,  and  get 

(Sx'^  —4xy  +  5y''')-'(2x^-  —2xy—4y^). 


24  UNIVERSITY   ALGEBRA. 

Second,  we  remove  each  of  these  parentheses,  taking 
care  to  change  all  signs  within  the  second  parenthesis, 
and  get 

Third,  we  arrange  these  terms  so  that  similar  terms 
shall  come  together,  and  get 

Sx'^—2x^  —  4xj/+2xy+6y-i-4:j/^, 
Fourth,  we  unite  each  group  of  similar  terms  into  a 
single  term,  and  get 

x^-'2xj/+dy^, 
and  this  is  the  simplest  form  possible  for  the  difference  of 
the  two  given  expressions. 

62.  It  is  easy  to  see  that  we  can  find  the  difference  of 
any  two  expressions,  whatever  the  expressions  may  be, 
in  a  manner  similar  to  that  just  pursued.  To  do  this  we 
enclose  each  expression  in  a  parenthesis,  write  the  min- 
uend first  and  the  subtrahend  second,  with  a  minus  sign 
between  them,  and  then  remove  parentheses.  Next  we 
arrange  the  terms  of  the  expression  thus  found  so  that 
similar  terms  shall  come  together.  Finally,  we  unite 
each  group  of  similar  terms  into  a  single  term. 

EXAMPLE. 

1.  From  x^+y^  take  x^-^y^. 

2.  From  x+aSd^  take  —x—Sa+b^. 

3.  From  a—b+c—d  take  a+b'-c+d, 

4.  From  2;i8+3a»— r»— ^«  take  «»— a»+r«— 25«. 

5.  From  a^+2ab+b^  take  a'^—2ab+b^. 

6.  What  must  be  added  to  r^+s^  +  tl  to  produce  3? 

7.  What  must  be  subtracted  from  abc^  to  produce  m'\-r^ 

8.  What  must  ^ab  be  subtracted  from  to  produce  —abl 


SUBTRACTION.  25 

9.  From  ahc^  —  lab'^c+Za'^bc  take  Zabc^ +1ab'^c+a'^bc. 

10.  From  the  sum  of  a^  +  ^^  and  —^ab  subtract  the  sum 
of  a^ — b'^  and  3^^. 

11.  From  x^-{-ax'^-\-a'^^x+d^  subtract  ^ax'^-'O^x^  and 
from  this  difference  subtract  ^ax'^—a'^x, 

63.  Arrangement  of  Work  in  Subtraction.  In  find- 
ing the  difference  of  two  expressions  by  the  method 
already  learned,  we  place  each  expression  in  a  parenthesis, 
write  the  subtrahend  after  the  minuend  with  a  minus  sign 
between  them,  and  then  remove  the  parentheses.  But, 
evidently,  it  comes  to  the  same  thing  if,  instead  of 
writing  the  subtrahend  with  all  its  signs  changed  after 
the  minuend,  we  write  the  subtrahend  with  all  its  signs 
changed  under  the  minuend,  with  similar  terms  of  the 
minuend  and  subtrahend  in  the  same  vertical  column, 
and  then  unite  similar  terms  exactly  as  in  addition. 

For  example,  if  we  wish  to  subtract  2^2__4^2__g  irova 
9(3^2_{_3^2_7^  ^g  arrange  the  work  thus: 

Minuend,  9^2^3^2__7 

Subtrahend  with  signs  changed,  — 2^2_j_4^2_|_g 
Remainder,  1  a'^  ^1  b''- —  \ 

The  signs  of  the  subtrahend  need  not  actually  be 
changed  if  the  student  will  imagine  them  changed  as  he 
proceeds  in  the  work. 

KXAMPi^KS. 
I.  2. 

From      15a— 73+3^— 7^— 8(?      7^— 2>/—  z+A+a 
take         10^  + 7^—3^4- 4^+ 4g        x^  y^b2—2       +n 

3.  From  4:x'^-\-2xy+^y'^'  take  x'^-'Xy-\-2y^. 

4.  From  1x^-—^x—l  take  bx'^—^x^-Z. 


26  UNIVERSITY   ALGEBRA. 

5.  From  Sx^—2x'^  +  Sx—4  take  x^—ix'^-Sx+l. 

6.  From  6ay—5xy+2a^x^  take  4:xy—Say—a'^x'^, 

7.  From  ia+^d—^'-9d+\  take  |«— f^— f^+i^— «. 

8.  From  4:ady^—5axy+2a^d'^  take  a2^2_^^^_3^^^2^ 

9.  From  2x+lla+10d—5c'-2S  take  2^-10+5^—33. 

10.  From  x^  +  Sxy—y^  +j/2:—2j/^  take  jr^  +  2;r^  +  5;r5r 
— 3y2_2^^ 

11.  From  6x'^  +  7xy—5j/^—12xyz—8yjs  take  8;r|/— Ty^f 

12.  From  4r^+62;^2_26w— 23;^2  ^ake  9;e2_2r^+21;;^ 
+  2n'^—Srs, 

INSERTION  AND   R^MOVAI,   OF  PARBNTHE;s:^S. 

64.  Parenthesis  within  a  Parenthesis.  It  may 
sometimes  happen  that  an  expression  within  a  paren- 
thesis is  itself  an  expression  which  contains  a  paren- 
thesis, so  we  have  a  parenthesis  within  a  parenthesis. 
Indeed,  we  may  have  several  parentheses  one  within 
another. 

These  complicated  expressions  present  no  difficulty, 
for  we  can  take  the  parentheses  one  at  a  time,  and  if  we 
know  how  to  remove  one,  we  may  do  this  and  then 
remove  another,  and  so  on  until  all  are  removed.  For 
example,  if  we  wish  to  remove  the  parentheses  from 

we  begin  by  removing  the  inner  parenthesis  first  and 
write  the  expression  in  the  form 

a  +  (id—c+d+e), 
and  now  by  removing  the  remaining  parenthesis  we  write 
the  result  in  the' final  form 

a-\-d—c-\-d-\-e. 


SUBTRACTION.  2/ 

65.  Until  some  skill  has  been  attained  it  is  usually 
best  for  the  beginner  to  remove  the  innermost  paren- 
thesis first,  and  then  the  innermost  parenthesis  of  all  that 
remains,  and  so  on  until  all,  or  as  many  as  may  be  de- 
sired are  removed. 

KXAMPLKS. 

Remove  the  parentheses  from  the  following  expressions: 

I.   a^-ib-{c+dy]. 

3.  5a3-(4^3_[-3(^2_j.^2)_4(^__2)])  +  2. 

4.  i_[i_(i«[i_(i_^)])]4.;^. 

5.  Enclose  the  last  three  terms  of  a-\- b—4:C—he'^  +Q>r^ 
—n^  —  16  within  a  parenthesis  preceded  by  a  +  sign. 

6.  Enclose  the  third  and  fourth  terms  of  a~\-d—4c 
-f5^2+6r^  —  ;^*  — 16  within  a  parenthesis  preceded  by  a 
—  sign,  and  the  fifth,  sixth,  and  seventh  terms  of  the  same 
expression  in  another  parenthesis  preceded  by  a  —  sign. 

7.  Fill   out   the   blank    parenthesis   in   the   equation 

8.  Fill   out  the   blank    parenthesis  in   the  equation 


CHAPTER  IV. 

MUIvTlPIylCATlON. 

66.  Extended  Definition.  To  Multiply  one  number 
by  another,  we  do  to  the  first  what  is  done  to  unity  to 
produce  the  second.*  The  number  that  is  multiplied  is 
called  the  Multiplicand,  and  the  number  multiplied  by 
is  called  the  Multiplier. 

In  the  present  chapter  we  will  understand  the  sign  X 
to  mean  ''multiplied  bf  (not  ''times'"^,  so  that  ay.h  means 
that  a  is  to  be  multiplied  by  b,  and  axbxc  means  that  a 
is  to  be  multiplied  by  b  and  the  result  multiplied  by  c. 

The  original  meaning  of  muhiplication  in  Arithmetic  is  that  of 
repeated  addition,  and,  with  this  meaning  in  mind,  we  would  define 
multiplication  to  be  the  taking  of  one  number  as  many  times  as  there 
are  units  in  another.  Thus,  3  multiplied  by  5  means  3+3+34-3+3, 
and  I  multiplied  by  5  means  f+f+l+f+f.  As  soon,  however,  as  the 
multiplier  is  a  fraction,  it  is  found  that  this  meaning  of  multiplication 
does  not  apply;  for  while  3  can  be  repeated  5  times,  yet  3  cannot  be 
repeated  \  a  time,  nor  can  |  be  repeated  f  of  a  time.  Now,  although 
the  operation  of  multiplying  |  by  f  cannot  be  looked  upon  as  repeated 
addition,  yet  this  operation  does  occur  in  Arithmetic,  and  is  called 
multiplication.  It  is  plain,  therefore,  that  the  word  is  used  with 
some  other  meaning  than  that  originally  given  it,  which  new  meaning 
we  have  stated  above. 

67.  Illustrations.  To  multiply  3  by  5  we  must  do  to 
3  what  is  done  to  unity  to  make  5.     But 

6=1  +  1  +  1  +  1  +  1; 
whence  3x5=3+3+3+3+3,  or  15. 

♦Boset  Algebra  Elementaire,  Charles  Smith's  Algebras. 


MULTIPLICATION.  29 

To  multiply  f  by  5  we  must  do  to  f  what  is  done  10 
unity  to  make  5.     But 

5=1  +  1  +  1  +  1  +  1; 
hence  2  x5=|+|+|+|+|,  or -i/. 

To  multiply  5  by  f  we  must  do  to  5  what  is  done  to 
unity  to  make  f ;  that  is,  we  must  divide  5  into  4  equal 
parts,  giving  f ,  and  take  one  of  these  parts  3  times. 
Hence,  5  X  f =|  X  3=-^^. 

To  multiply  f  by  f  we  must  do  to  f  what  is  done  to 
unity  to  make  f ;  that  is,  we  must  divide  f  into  5  equal 
parts,   giving  y^-,  and  take  one  of  these  parts  4  times. 

2  4        2,     2x4 

Hence,  oXf=^s — p^^—^ — i-* 

3  5     3x5  3x5 

In  the  general  case  of  the  product  of  two  fractions  we 

would  find, 

n    s_nXs  r.^-, 

68.  Law  of  Signs  in  Multiplication.  The  definition 
of  multiplication  given  above  is  sufficiently  broad  to  in- 
clude the  case  of  multiplication  by  negative  numbers. 

To  multiply  —6  by  3  we  must  do  to  ~6  what  is  done 
to  unity  to  produce  3.     But 

3  is  1  +  1  +  1; 
hence,  —6x3  is  —6—6—6,  or  —18. 

To  multiply  6  by  —  3  we  must  do  to  6  what  is  done  to 
unity  to  produce  —3.     But 

-3  is -(1  +  1  +  1); 
hence,  6x —3  is —(6  +  6  +  6),  or  —18. 

To  multiply  —6  by  —3  we  must  do  to  —6  what  is  done 
to  unity  to  produce  —3.     But 

-3  is -(1  +  1  +  1); 
hence,  — 6x  — 3  is  —(—6-6—6),  or  +18. 


[2] 


30  UNIVERSITY    ALGEBRA. 

Likewise  it  may  be  seen  that  the  definition  includes  the 
case  where  the  multiplier  is  a  negative  fraction.  Hence, 
if  a  and  b  stand  for  any  two  numbers,  positive  or  nega- 
tive, integral  or  fractional,  we  have 

(+a)X(+&)  =  +a&  (1)1 

(-a)X(+&)  =  -a&  (2) 

(+a)X(-&):=-a&  (3) 

(-a)X(-&)=+a&  (4) 

These  equations  express  the  Law  of  Signs  in  mul- 
tiplication, which  is  often  stated  thus :  Like  signs  give 
+  and  unlike  give  — . 

69.  Commutative  Law  of  Multiplication.  From 
the  law  of  signs  we  observe  that  the  sign  of  a  product 
will  be  the  same  no  matter  in  what  order  the  factors  are 
multiplied  together.  Thus,  the  product  (4-<2)  x  ir-U)  has 
the  same  sign  as  the  product  (— <^)x(  +  a).  We  can 
show  a  similar  truth  respecting  the  absolute  value  of  the 
product. 

First,  suppose  both  multiplicand  and  multiplier  are 
positive  whole  numbers:  we  wish  to  prove  axb=bxa. 

We  may  write  down  b  rows  of  units  with  a  units  in 
each  row,  thus : 

1     1     1     1     1  .  .  .  « columns. 
11111... 
11111... 
11111... 


b  rows. 
Then,  since  there  are  a  units  in  1  row,  in  b  rows  there 
are  axb  units.  But  since  there  are  b  units  in  1  column, 
in  a  columns  there  are  bxa  units.  Of  course  the  number 
of  units  is,  in  either  case,  the  same,  so  axb=bxa,  if  a 
and  b  are  positive  whole  numbers. 


iM  ULTIPLICATION.  3  I 

Second,  suppose  multiplicand  and  multiplier  are  pes- 

7Z  S 

itive  fractions,  say  —  and  -.     In  Art.  67  we  found 
r  t 

n     s     nxs 
y/_—— . 

r     t     rxt 

Similarly  we  would  find 

s     n     sxn 

-x-=- 

t     r     txr 

But  we  have  just  shown  that  nXs=sXn^  and  also  that 

rXt=tXr,  hence 

71     s     s     n 
-X-=-X  — 

r     t     t     r 

Hence,  for  all  positive  values  of  a  and  h^  integral  or 
fractional,  a  x  b=^  bxa. 

But,  from  the  law  of  signs,  the  sign  of  the  product  does 
not  depend  upon  the  order  of  the  factors ;  therefore,  for 
all  values  of  a  and  b,  positive  or  negative,  integral  or 
fractional,  axh=bxa,  [3] 

That  is,  zn  the  product  of  any  two  numbers,  we  may  take 
the  factors  in  either  order.  This  is  called  the  Commu- 
tative Law  of  Multiplication. 


70.  Case  of  Three  or  More  Factors.  Suppose  each 
of  the  units  in  the  rectangular  arrangement  in  the  pre- 
vious article  be  replaced  by  a  number  whose  value  is 
represented  by  a.  I^et  there  be  b  columns  and  c  rows,  thus: 

a     a     a     a     a  ,  .  ,  ^columns. 

a     a     a     a     a  ,  ,  , 

a     a     a     a     a  .  ,  , 

a     a     a     a     a  ,  ,  » 


c  rows. 
Now  the  total  number  of  units  in  the  arrangement  may 
be  estimated  in  different  ways.    The  number  in  each  row 


32  UNIVERSITY   ALGEBRA. 

is  axd,  and  hence  the  number  in  c  rows  is  axdxc.  The 
number  in  each  column  is  aXc,  and  hence  the  number  in 
d  columns  is  aXcxb, 

Therefore,  axbxc=axcxb,  (1) 

Since  axb=bxa,  axbxc=bxaxc,  (2) 

Putting  bxaiox  axb  and  cXa  ior  ax c  in  (1), 

bxaxc=cxaxb.  (3) 

From  (1)  we  see  that  a  product  remains  the  same  if  the 
second  and  third  factors  be  interchanged,  from  (2)  we  see 
that  a  product  remains  the  same  if  the  first  and  second 
factors  be  interchanged,  and  from  (3)  we  see  that  the 
same  is  true  if  the  first  and  third  be  interchanged. 

Whence  we  write, 

ax:f>xc=axcxh=^bxaxc=hxcxa=cxaxh=cxl>xa    [4] 

This  is  the  commutative  law  for  three  factors.  The 
law  may  be  shown  to  hold  for  fractional  and  negative 
factors  exactly  as  in  the  last  article. 

71.  Associative  Law  of  Multiplication.  The  num- 
ber of  a's  in  the  rectangle  in  the  last  article  is  be.  Whence 
the  total  number  of  units  is  ax{bxc).  By  the  last  article 
the  total  number  of  units  equals  axbxc,  ox  (axb^Xc, 
whence  we  write, 

aXhXc={aXh)Xc=aX{hXc)  [5] 

Thus,  3x7x5  is  the  same  as  21  x  5  or  3  X  35.  This 
fact,  that  multiplication  may  be  performed  by  grouping 
the  factors  in  diJfferent  ways,  is  called  the  Associative 
Lavv  of  Multiplication. 

By  means  of  [4]  and  [5]  the  commutative  and  associa- 
tive laws  can  easily  be  extended  to  the  product  of  any 
Qumber  of  factors,  negative  or  fractional. 


MULTIPLICATION.  33 

72.  Index  Law  of  Multiplication.  From  the  defi- 
nition of  a  power  of  a  number  we  have  a^  x  a^  =  aaXaaa 
=  ^5_^2-t-3  Also  a^  Xa^  =  aaaXaaaaa=a^=a^^^,  In 
general : 

a''=aaaa  ...  to  «  factors 
ar=aaaa  ...  to  r  factors 
a'*xa''=  {aaa  ...  to  ^  factors)  X  (aaa  ...  to  r  factors) 

=aaaa  ,  ,  ,  to  (n  +  r)  factors      by  associative  law, 
=<2""^''  by  definition  of  an  exponent. 

What  we  have  now  proved  may  be  stated  in  the  fol- 
lowing words : 

The  product  of  two  powers  of  the  same  number  is  equal 
to  that  number  with  an  exponent  equal  to  the  sum  of  the 
exponents  of  the  two  factors, 

73.  Formulas.  Statements  like  the  above  expressed  in 
the  symbolic  language  of  Algebra  are  called  Formulas. 
Thus :  a«  X  a^=<x"+''  [6] 
is  a  formula.    So  also  are  [1],  [2],  [3],  [4],  and  [5]  above. 

PRODUCT   OF   MONOMIALS. 

74.  The  following  illustrate  the  laws  of  multiplication 
thus  far  proved : 

(1)  Find  the  product  of  ^ab  by  hxy. 

(^^ab)  X  (bxy)  =  ^abbxyy         by  associative  law, 
=  ?>Oabxy,      by  commutative  law. 

(2)  Find  the  product  of  Zax"-  by  —2bx^. 

(^ax'^)  X  {-2bx^')=-(^ax'')(2bx^), 

by  law  of  signs, 
=  — 6^^Jt^^ 
by  associative,  commutative,  and  index  laws. 
Thus  the  product  of  any  two  monomials  can  be  obtained 
by  means  of  the  four  laws  already  set  forth. 


34  UNIVERSITY    ALGEBRA. 

KXAMPI^KS. 
Multiply 

1.  ^''x^  by  hh^p^x'^,  5.  1^-^hx  by  -5^3^^ 

2.  -8/^2  by  i^3^3^  6,  -33^^  by  lOi^y . 

3.  60;i:j/2  by  .05^3jK.  7.  (^+jk)'^  by  4(jr+jj/)^ 

4.  -20^2^  by  -.3^2^.  8.  3(;r--5)2  by  ^(^-5)*. 

PRODUCT   OF   A   POI^YNOMIAI.  AND   A   MONOMIAL. 

75.  Distributive  Law  of  Multiplication.  We  wish 
to  prove  {a-{-U)n=^an-\-bn,  where  a,  by  and  n  stand  for 
any  numbers  whatever. 

First,  we  let  nhe  a,  positive  whole  number.     Then 
(a+d')n=(a  +  d)  +  Ca-\-d)-i-(a-^d)  +  .  .  .  to  ;z  terms, 

by  definition  of  multiplication, 
=a+d+a+d+a  +  d+.  .  . 

by  associative  law  of  addition, 
'=a+a+.  .  to  w  terms  +  ^4-^+.  .  to  n  terms, 

by  commutative  law  of  addition, 

sszan+dn,  (1) 

by  definition  of  multiplication. 

Second,  let  ^  be  a  positive  fraction,  represented  by  -• 

Then,  if  division  be  defined  as  the  process  of  undoing 
multiplication ,         (a-i-  d)-i-r=a-^r-j-  d-^r,  (2) 

if  r  be  a  positive  whole  number;  for  if  we  multiply  either 
side  of  the  equation  by  r  we  shall  obtain  a-j-d. 

But  (/i+^h=—j-xs, 

by  definition  of  multiplication, 

=  (a--/+^-T-0^,  by  (2), 
=a-r-/X^+<^-r-/X^,  by  (1),    . 

=ax^+^x^  (3) 

by  definition  of  multiplication. 


MULTIPLICATION.  3  5 

Third,  let  —r  represent  any  negative  multiplier,  inte- 
gral or  fractional.     Then,  from  (1)  and  (3), 

{a-^U)r—ar-\-br,  (4) 

whence  —{a  +  b)  r=  —  {ar-\-bf),  (5) 

—{cL-\-by-=—ar—br,  (6) 

by  removing  parenthesis. 

Now,  by  the  law  of  signs,  the  members  of  (6)  may  be 

written  as  follows : 

(«  +  ^)(-^)=«(-^)  +  ^(-r).  (7) 

Therefore,  for  all  values  of  n, 

{a+b)n=an-{-bn.  [8] 

76.  More  than  Two  Summands.  In  (a+b)n  sup- 
pose a=x-{-y.     The  equation  {a-\-  b)n=an-[-  bn  becomes 

(x-{-y+b')n=^(x-\-y)n-{-b7i^xn-\-yn-{-bn, 
Likewise, 

['OC-^y-{-z-\-w-{-.  .  .)n=xn-{-yn+zn+wn+.  .  .  [9] 

That  is,  ike  product  of  any  polynomial  by  a  monomial 
equals  the  sum  of  all  the  terms  found  by  multiplying  each 
term  of  the  polynomial  by  the  given  monomial. 

The  above  is  called  the  Distributive  Law  of  Multi- 
plication. 

77.  The  following  examples  illustrate  the  principles 
of  multiplication  we  have  thus  far  proved : 

(1)  Find   the   product   of   {Zax-\-^.bx'^—^y^   by  Zxy, 
(Sax+4bx'^—4y')xSxy==dax'^y-hl2bx^y—12xy^. 

(2)  Find   the   product   of  6x^—4x^+8x  by  ~^ax^, 
{^x^—Ax'^-\-Sx'){-'\ax'^)=—?>ax^-\-2ax^—4:ax^, 

78.  Arrangement  of  Work.  The  following  illustra- 
tions will  be  sufl&cient  to  explain  the  usual  arrangement 
of  work  when  the  product  of  a  polynomial  by  a  monomial 
is  sought: 


36  UNIVERSITY   ALGEBRA. 

(1)  Multiply  a-\-b—c  by  5. 

Multiplicand,     a+   b—  c 
multiplier,  5 

product,  ha-\-hb—hc 

(2)  Multiply  :r-3;r2  +  2;t:3  by  -5^. 

Multiplicand,         x  --  Zx'^+  2x^ 

multiplier,        —hx 

product,  —bx'^ -{-Ibx^ —  lOx^ 

(3)  Multiply  ^ay'^  by  la'^y'^—^ay. 

Since  multiplication  may  be  perforTned  in  a7iy  order,  we 
arrange  this  just  as  though  ^ay'^  were  the  multiplier. 

KXAMPi^KS. 

I.  2. 

Multiply       2x—5y+B  Multiply         ba'^  —  Aa-^Q 

by  11  by  —4:ax 

Multiply 

3.  2x—4y-}-2  by  Sx. 

4.  a^d  by  4:a+2ac^—Sd^  —  l, 

5.  Sab-\-4a'^b—5cb-{-6  by  —Aa^d\ 

6.  6a4-4^2_|_2^^3_3^2  by  -_i.^2^^^ 

Simplify  each  of  the  following  expressions : 

4(4^-73)=16«-283;        _7(2^_5^)  =  _i4^-|-35^. 
Therefore,  4(4^-73)-7(2a-5<5)=:16«-28^-14«+35^. 

Combining  similar  terms,  =2^+7^. 

8.  a(ia+b'-c)—b(a—b+c)-hc(a-Jtb), 

9.  3«^(«— ^)— ^(r(2^— 3a)-f6a32  — ^2(3^_|.2^). 
10.  ab'^(iax—bx+eax^—5bx^)4:a^b. 

IX.  i;r)/2(^3— ;ry2+9;i:2jj/— 7>/^)8^^ 


MULTIPLICATION.  37 

PRODUCT   OF   A   POI.YNOMIAI,   BY   A   POI.YNOMIAI,. 

79.  Law  of  Product  of  two  Polynomials.  In  the 
equation,  (a+d+c+,  .  ,)n=an+dn+cn  +  ,  .  .  (1) 
put  n=x+y;  then  we  have, 

(a  +  d+c+.  .  ,)(xA-y)=aix+y)  +  d(ix+y)  +  cCx+y)  +  ,  .  . 
by  commutative  law,  ={x-{-y)a-\-(x+y)b+(x+y)c+,  .  , 
by  distributive  law,     ■=xa-\-xb+xc+ .  ,  . 

Also,  by  putting  (x-}-y+z+.  .  .)  for  n  in  (1),  we  get 
(a  +  b+c+.  .  .)(x+y+2+,  .  ,')=xa-\-xb-{-xc+,  .  . 

+ya-j-j/d+yc+.  .  . 

-hs^a-^-  ^d-{-  2c-{-.  .  .,  etc. 
Stating  this  in  words,  fke  product  of  a  polynomial  by  a 
polynomial  is  the  aggregate  obtained  by  placing  one  poly- 
nomial as  a  factor  in  each  term  of  the  other.  Or,  in  other 
words :  The  product  of  one  polynomial  by  another  is  the 
sum.  of  all  the  terms  found  by  multiplying  each  term  of  one 
polynomial  by  each  term  of  the  other  polynomial, 

80.  We  illustrate  this  by  a  few  examples : 

(1)  Multiply  ;r— 4  by  ^+9. 

Placing  the  second  polynomial  as  a  factor  in  each  term 
of  the  first  polynomial,  we  have 

;r(;r+9)-4(;i:+9). 
Multiplying  by  x  and  —4,  we  obtain 
x'^-\-^x—\x—Z^. 
Uniting  similar  terms,  we  have  the  required  product, 

(2)  Multiply  3^2^ 2^~9  by  2a^-6^. 

Placing  the  second  polynomial  as  a  factor  in  each  term 
of  the  first  polynomial,  we  have 

3ij2(2a^— 6^)  +  2a(2«^— 6<^)--9(2^^-6^). 


38  UNIVERSITY    ALGEBRA. 

Multiplying  by  3^^,  la,  and  —9,  we  obtain 

Uniting  similar  terms,  we  have  the  required  product, 
^a^b-\Wb-?>^ab-\-hU. 

81.  Arrangement  of  Work.  Let  us  go  through  the 
work  of  multiplying  the  polynomial  2<22  — 3^  — 3  by 
3^2— 2<2+2.  We  first  place  the  second  polynomial  as  a 
factor  in  each  term  of  the  first  polynomial,  and  obtain 
2a2(3a2-2«  +  2)-3^(3^2_2a+2)-3(3^2_2^_^2).  (1) 
Then  multiplying  the  expressions  in  the  parentheses  by 
2^^^  — 3<3:,  and  —3,  respectively,  we  obtain 

Now,  uniting  similar  terms,  we  get  the  required  product, 

^a^—\Za^+a''-^,  (3) 

This  work  may  be  arranged  in  a  more  convenient  form : 


Multiplicand, 

2^2—  3^  —3 

(4) 

multiplier, 

3^2_  2a  +2 

(5) 

first  partial  product, 

6a4_  9a3_9^2 

(6) 

second  partial  product, 

-  4a=^  +  6a2_|.e^ 

(7) 

third  partial  product. 

4^2_e^_6 

(8) 

product. 

6a4_i3^34.  ^2         _e 

(9) 

which  is  the  usual  arrangement  of  work  in  the  multipli- 
cation of  two  polynomials. 

82.  The  following  examples  will  tend  to  show  the 
advantage  of  arranging  the  work  of  multiplication  in  the 
way  explained : 

x-\-y  a — b  x'^  +xy-\-y^ 

x-\-y  a-\-b  x—y 

x'^ -\-  xy  a^  —  ab  x^ -\- x^y -\- xy"^ 

xy-^y^  ab—b^  —x'^y—xy'^—y^ 

x'^-\-2xy-\-y'^  a^  —b'^  x^  — jK* 


MULTIPLICATION.  39 

' '  The  student  should  observe  that,  with  the  view  of  readily  bring- 
ing similar  terms  of  the  product  into  the  same  column,  the  terms  of 
the  multiplicand  and  the  multiplier  are  arranged  in  a  certain  order. 
We  fix  on  some  letter  which  occurs  in  many  of  the  terms  and  arrange 
the  terms  according  to  the  potvers  of  that  letter.  Thus,  taking  the 
last  example,  we  fix  on  the  letter  X',  we  put  first  in  the  multiplicand 
the  term  x^,  which  contains  the  highest  power  of  x,  namely  the 
second  power;  next  we  put  the  term  xy  which  contains  the  next  power 
of  X,  namely  the  first;  and  last  we  put  the  term  y^  which  does  not 
contain  x  at  all.  The  multiplicand  is  then  said  to  be  arranged 
according  to  descending  powers  of  x.  We  arrange  the  multiplier  in 
the  same  way. 

"We  might  also  have  arranged  both  multiplicand  and  multiplier 
in  reverse  order,  in  which  case  they  would  be  arranged  according  ta 
as cendiftg powers  of  x.  It  is  of  no  consequence  which  order  we  adopt, 
but  we  should  take  the  same  order  for  the  multiplicand  and  the  mul- 
tiplier."—  Todhunter's  Algebra  for  Beginners. 

EXAMPI.KS. 

Multiply  the  following  expressions : 

1.  x—lZ  by  x—14:.  3.  ^x'^  +  2y'^  hy^x'^  —  ^y'^ 

2.  x+1^  by  .r— 20.  4.  x'^—xy+y'^  by  x+y. 

5.  2.r2-3+5.;t:8  by  Qx-S-^Ax^. 
Arranging  according  to  the  descending  powers  of  x,  we  obtain 
5x^-\.2x»-3 
4x^-{-Qx  -8 


20a:«+8a;6  -12^8 

+30^*4-12^3  __i8^ 

-40^8-16.y«  +24 

20^«+8^5_{.3o,c4_40;»:8_i6;»:8_lg;P4-24 

6.  ab^+3aH'-2aH^  by  2a^-ad-5d^. 

7.  x^'-5ax—2a^  by  x^+2ax+Sa^. 

8.  7x'^+y'^—Sxy  by  2x^+y—x. 

9.  2x^—4x'^—4x--l  by  2x^—4^'^'-Ax-'l. 
10.  5.r— 7.;i;2_^.r3  +  l  by  l+2x'^-'4cX. 


40  UNIVERSITY    ALGEBRA. 

II.  a^-\-b'^-\-c'^  —  hc—ca—ab  by  a-\-b-\-c, 

13.  x''"  -{-y"^  -^ z'^  -{■  xy—y2-\-xz  by  x—y-\-z. 

14.  ^^2-1^-1  by  1^2  ^1^-3. 

15.  2^_|^4-5^  by  \m\-\r-^p, 

SPBCIAI,    PRODUCTS. 

83.  Product  of  Sum   and   Difference.     By  actual 

multiplication  we  find 

(a+&)(a-5)=:a2-&2.  [10] 

Since  a  and  b  may  stand  for  any  numbers,  we  may  say: 
The  product  of  the  sum  and  the  difference  of  any  two 
numbers  equals  the  difference  of  their  squares. 

84.  Square  of  a  Binomial.  By  actual  multiplication 
we  find  (<:e+&)2=as+2a&+&S  [11] 

{a-b}^=a^-2ab+b^.  [12] 

Since  a  and  b  may  stand  for  any  numbers,  we  may  say: 

The  square  of  the  sum  of  two  numbers  is  equal  to  the 
square  of  the  first,  plus  twice  the  product  of  the  twOy  plus 
the  square  of  the  second. 

The  square  of  the  difference  of  two  numbers  is  equal  to 
the  square  of  the  first,  minus  twice  the  product  of  the  two^ 
plus  the  square  of  the  second. 

85.  Products  of  the  Form  {x-^a^{x-^b).  By  actual 
multiplication  we  find 

(ir-ha)(iZJ+&)=a5»+(a+&)ir+a&.  [13] 

That  is,  the  product  of  any  two  binomials  in  which  the 

first  terms  are  alike  is  equal  to  the  square  of  the  first  term 

plus  the  first  term  with  a  coefficient  equal  to  the  sum  of  the 

second  terms,  plus  the  product  of  the  second  terms. 


MULTIPLICATION.  4 1 

Of  course  due  attention  must  be  paid  to  the  signs  of 
the  terms  in  writing  out  such  products.     Thus : 

KXAMPI^ES. 

Write  down  the  square  of  the  following  binomials : 


I. 

ad+c. 

5. 

A:x—^ab, 

9- 

m^-r'-s\ 

2. 

c—ab. 

6. 

oab—Sac, 

10. 

(al>^y  +  i2xy. 

3. 

x-y. 

7- 

i-i- 

II. 

(2xy+2x-'. 

4. 

lax^b. 

8. 

100-1. 

12. 

a+a. 

Write  by  inspection  each  of  the  following  products : 

13.  (ix-\-a)(ix—d).    18.  (x+yX^—y')'    23.  (ix-\-2y)(x—2y) 

14.  (x+lX^-1).    19.  (b''+cXb''-c).  24.  (jr2+4)(^2_4) 

15.  (a  +  25)(«-4).  20.  (a-2)(a-3).     25.  (a'^-l')(a'^-j-2) 

16.  (^-^)(a  +  3).    21.  (a  +  2)(a  +  3).     26.  Ca  +  bc)(a-cd) 

17.  (a  +  10)(a-5).  22.  (^— 6)(«+5).     27.  (a+^^)(^  +  ^^') 

28.  (2^-^)(2a  +  3^).  31.  (a^^+^2)(^^2_2). 

29.  (2«— <^2)(2a~^*'^).         32.  (mn  +  o^^mn—A), 

30.  (.;i:H-3j/2')(^— 2z^z;).       33.  (;r2jj/+^jj/2)(^2^_;rK2)^ 

86.  Generalized  Law  of  Distribution.    I^t  us  con- 
sider the  product 

(^1+^2+^3  +  -    .    .+^(^1+^2+^3+.    .    .+^r)  (1) 

where  ^j,  ^3,  ^a  .  .  .,  ^1,  ^2)  <^3  •  .  .»  stand  for  any  num- 
bers whatever.  We  have  supposed  that  there  are  n  terms 
in  the  first  parenthesis  and  r  terms  in  the  second  paren- 
thesis. By  the  distributive  law  the  product  will  consist  of 
the  sum  of  all  the  partial  products  found  by  multiplying 


42  UNIVERSITY   ALGEBRA. 

each  term  in  one  parenthesis  by  each  term  in  the  other 
parenthesis.     Thus,  the  product  may  be  written, 

'\-a^hr^-a^br-\-a^br-\-.  .  .-\-aJ)r  (2) 

As  we  have  written  the  result,  there  are  n  columns  of 
terms  with  r  terms  in  each  column.    Therefore,  the  total 
number  of  terms  in  the  product  is  nr. 
Next  consider  the  product 

(«^4-«,  +  ^3+.   .    ^.)(^,  +  ^,+  ^3+.   .    ^.)(^,  +  ^,  +  ^3+.   .   6^       (3) 

where  there  n  terms  in  the  first  parenthesis,  r  in  the 
second,  ^  in  the  third,  and  so  on. 
This  product  may  be  written 

(«x^x+^A+-  .+^A+^A-  .)(^.+^.+^3+-  •^^)'     W 

where  the  first  parenthesis  is  suppose  to  contain  the  nr 
terms  written  in  (2)  above.  We  now  have  a  parenthesis 
containing  nr  terms  to  be  multiplied  by  a  parenthesis 
containing  ^  terms.  This  will  give  nrxs  or  nrs  terms, 
which  may  be  written  out  just  as  in  the  case  of  (1) 
above : 

^i  Vi  +  •  •  +  ^^  Vi'  +  ^i  Vi  +  •  •  +  ^;.  Vi'  +  •  • )  +  aj)^c^  + . .  +  a„brC^ 

ab^Cs-\- . .  +  a,fi^Cs,  +  aj?^:^ + . .  +  a,fi^c,,  +  •  • ,  +  ab^c^^- . .  +  a,Jb^^ 

By  a  continuation  of  this  process  we  could  evidently 
write  out  the  product  of  any  number  of  parentheses. 
Whence  we  say: 

The  product  of  k  parentheses  is  the  aggregate  of  Aix,  the 
possible  partial  products  which  can  be  made  by  multiplying 
together  k  terms,  of  which  one  and  only  one  must  be  taken 
from  each  parenthesis. 


MULTIPLICATION.  43 

If  the  number  of  terms  in  the  differe7it  parentheses  be 
n,  r,  s,  t,  .  ,  ,  respectively ^  the  total  number  of  terms  in  the 
product  will  be  nrst  .  .  . 

The  student  will  observe  that  the  above  is  merely  a 
general  statement  of  the  distributive  law.  The  state- 
ment of  the  law  in  this  form  is  of  the  greatest  import- 
ance. The  following  examples  will  tend  to  make  its 
application  clear: 

(1)  Write  out  {a  +  b')(ic+d)(ie+f). 

Here  we  write  ace  +  bee + ade + bde  -\-  acf-\-  bcf+  adf+  bdf 
for  the  product  of  the  three  parentheses  is  the  aggregate 
of  all  the  possible  partial  products  which  can  be  made  by- 
multiplying  together  three  terms,  of  which  one  and  only 
one  must  be  taken  from  each  parenthesis.  The  number 
of  terms  in  the  result  is  2  x  2  x  2  or  8. 

(2)  Writeout  («  +  ^)(^+0(^+^)• 
This  gives  abc-\-b'^c-\-ac'^-\- bc'^-\- a'^b+b'^a  +  a'^c+ bca. 

Here  the  first  and  last  terms  are  alike,  so  we  write: 
2abc^b''c-{-ac'^^bc'^+a^b-^b'^a-\-a'^c, 

(3)  Write  out  {a  +  /^)  (^  +  b)  (,a  +  b). 

The  possible  products  of  three  factors  each,  which  can 
be  made  from  the  two  letters  a  and  b  are  a^ ,  a'^b,  ab'^ , 
and  b^.  Thus,  while  the  above  product  may  be  written 
as  8  terms,  it  can  evidently  contain  but  4  distinct  terms. 
We  can  readily  determine  what  terms  are  repeated.  The 
term  a^  can  evidently  occur  but  once,  since  a  can  be 
taken  from  each  of  the  three  parentheses  in  but  one  way. 
The  same  is  true  of  b^ .  The  term  a'^b,  however,  may  be 
made  in  three,  ways:  first,  by  taking  b  from  the  third 
parenthesis  and  a  from  each  of  the  other  two ;  second,  by 
taking  b  from  the  second  parenthesis  and  a  from  each  of 
the  other  two ;  third,  by  taking  b  from  the  first  paren- 


44  UNIVERSITY   ALGEBRA. 

thesis  and  a  from  each  of  the  other  two.  lyikewise,  the 
term  ab'^  occurs  three  times.  So  we  write  the  above 
product,  a^-^-Za'^b-^Zab'^-^b^, 

87.  Type  Term.  All  terms  that  can  be  derived  from 
one  another  by  interchanges  among  the  letters  are  said 
to  be  terms  of  the  same  Type.  Thus,  a'^ ,  b'^ ,  etc.,  are 
of  the  same  type,  also  a^ ,  b^ ,  etc.,  also  a'^b,  a^c,  b'^a,  etc. 

The  number  of  terms  of  the  same  type  will  depend  on 
the  number  of  letters  we  have.  Thus,  if  we  have  the 
four  letters  a,  b,  c,  d,  there  are  four  terms  of  the  same 
type  as  <3^2,  six  terms  of  the  same  type  as  ab,  twelve  terms 
of  the  same  type  as  a'^b^  etc. 

88.  ^  and  U  Notation.  It  is  often  convenient  to 
indicate  the  sum  of  all  the  terms  of  the  same  type  by  an 
abbreviation.  This  is  done  by  prefixing  the  letter  ^, 
which  means  ''sum  of  all  terms  of  same  type  as''  or  ''sum 
of  all  such  terms  as, ' '  to  one  term  of  the  type.  Thus,  if  we 
have  the  letters  a,  b^  and  c,  the  symbol  '2a'^b  means 
a'^b-\-a'^C'\-b'^a-\-b'^c-\-c^a-\-c^b.  If  we  have  four  letters, 
a,  b,  c,  and  d,  the  symbol  ^a'^b  means  the  sum  of  more 
terms  than  those  just  written.  In  using  this  notation  it 
is  always  necessary  to  know  how  many  letters  are  to  be 
made  use  of,  but  this  is  generally  shown  by  the  context. 
Thus:  (a  +  by=^:Ea^  +  Z:EaH. 

The  product  of  several  factors  of  the  same  type  is  often 
indicated  in  a  similar  way,  namely,  b}^  prefixing  the 
letter  iT,  which  means  '  'product  of  all  factors  of  the  same 
type  as, "  or  '  'product  of  all  such  factors  as, ' '  to  one  factor 
of  the  type.  Thus,  if  we  have  the  letters.  <3^,  b,  and  c, 
na'^b=a'^bxa'^cy:b'^axb'^cxc^^axc'^b, 
n(a—b)={a  —  b)(^b'-c)(j:—a),  etc. 


MULTIPLICATION.  45 

89.  Cyclical  Order.  There  is  often  a  convenience  in 
arranging  the  letters  of  an  expression  in  a  peculiar  order. 
Thus,  in  the  expression  bc-{-ca-\-ab,  the  second  term  is 
made  from  the  first  by  changing  b  into  c  and  c  into  a,  the 
third  term  is  made  from  the  second  by  changing  c  into  a 
and  a  into  b.  A  similar  change  in  the  third  term  would 
give  the  first.  In  each  of  these  cases  one  term  is  said  to 
be  made  from  the  other  by  a  Cyclical  Change  or  by 
Advancing  the  Letters. 

The  expression  a(^b—cY  +  b(c—a)'^-\-c(a—bY  has  its 
letters  in  cyclical  order. 

90.  Number  of  Letters.  The  adjectives  Binary, 
Ternary,  Quaternary,  etc.,  are  useful  in  describing 
expressions  which  are  made  from  two,  three,  four,  etc., 
different  letters  respectively.  Thus:  a^b  is  a  binary 
term,  ab'^c'^  is  a  ternary  term,  xyzw  is  a  quaternary  term, 
and  so  on. 

91.  Product  of  Form  (^x+a){x^-b){x-\-c)  .  .  .  Sup- 
pose that  the  product  of  n  binomials,  in  which  the  first 
terms  are  alike,  is  sought:  say  {x-\-a){x-[-b')(^x-\-c)  .  .  . 
By  the  generalized  distributive  law  we  may  proceed  thus: 

All  the  ^'s  must  be  multiplied  together,  which  gives  x'' 
as  one  part  of  the  product.  Also,  a  must  be  multiplied 
by  the  ^'s  in  all  the  other  parentheses,  which  gives  ax''~'^ ; 
b  must  be  multiplied  by  the  x's  in  all  the  other  paren- 
theses, which  gives  bx*'~'^ ;  by  continuing  this  we  get  as 
as  another  part  of  the  product,  {a+b-]rc-\-.  .  .);r'*"~^. 

We  must  also  multiply  ahy  b  and  this  by  the  product 
[)f  the  ^'s  in  all  the  other  parentheses,  which  gives  abx''~'^ ; 
ike  wise  a\yY  c  and  this  by  the  x's  in  all  the  other  paren- 
theses; etc.;  by  continuing  this  we  get  another  part  of 
le  product,  {ab-\-ac+,  .  .-^-bc+bd-^- ,  .  ,^x''~'^. 


46  UNIVERSITY    ALGEBRA. 

In  like  manner  we  find  another  part  of  the  product, 
(abc-\-abd-\- .  .  .  +  bcd-\-bce-\-,  .  .')x"~^^ 
and  so  on. 

Finally,  all  the  numbers,  a,  b,  c,  etc.,  must  be  multi- 
plied together,  so  the  complete  product  may  be  written, 
(x-\-a){x-\-b')(x-{-c)  ...  to  ^  factors 

=x''-\-{a-^b+c+.  .  .)x''-^-\-{ab-^ac^,  .  .)x''-^ 

-\-{abc-\-abd-\-.  .  .)x''~^-\-.  .  .-{-abc  .  .  . 
As  we  have  written  this,  the  number  of  terms  is  n  +  1. 
If  the  parentheses  be  removed  the  number  of  terms  will 
be  2  X  2  X  2  ...  or  2". 

Using  the  ^  notation,  we  write : 

['3C-ra)[X-{-h)[0C-\-C)  .  .  .  to  n  factors 

=x''+2a.x''-^-}-2(ib,x''-^+2abc.x*'-^+. . ,+abc,. .  [14] 

92.  Square  of  a  Polynomial.  Consider  the  following 
important  product : 

(a  +  b+c+.  .  ,Xa+b+c+,  .  .) 
The  result  will  consist  of  nxn  or  n'^  terms,  if  n  repre- 
sents the  number  of  terms  in  each  parenthesis.  By  our 
generalized  distributive  law  we  must  take  a  from  one 
factor  with  a  from  the  other,  b  from  one  factor  with  b 
from  the  other,  etc.,  which  gives  a^ -j- b"^ -\- c^  + .  .  .  Also 
we  must  take  a  from  the  first  factor  with  b  from  the 
second,  and  b  from  the  first  with  a  from  the  second,  which 
gives  two  terms  equal  to  ab.  Likewise  for  the  terms  ac^ 
ad  .,.  be  ,,.;  so  that  we  say: 

(a+b+c+, .  ,y 

=  a^  +  b'^+c'^  +  .  .  ,+2ab  +  2ac+.  .  .+2bc+.  .  . 
or  {a-^b+c-\-. .  .)^=2a^-{-22ab.  [16] 

The  number  of  terms  in  the  result  is  nxn  or  n^,  of 
which  n  are  like  a^,  and  n^—n  like  ab.  The  number  of 
distinct  terms  \s  n-\-\{n'^  —n^  ox  \n{7i-\-V). 


MULTIPLICATION.  47 

KXAMPIvKS. 

Write  out  the  products  in  the  following  ten  examples : 
I.    (^b—c){c—a)(a—b). 

5.   {b-\-c)(ic-^a){a-\-b)(^b-'C){c-a){a-b). 

8.   (;r4-j^+-s'— ^— ^— ^)^ 

g.   (c+a  +  n-\-t—d+o—z  +  ty, 

11.  Prove (;t:+j/)*  =  2(;»;2+^2)(^^^^)2_(^2_y)2^ 

12.  Write  down  all  the  quaternary  products  of  the  three 
letters  a,  b,  c,  and  tell  how  many  different  types  they  fall 
into  and  how  many  products  there  are  in  each  class. 

13.  Do  the  same  for  the  binary  and  for  the  ternary 

products  of  the  four  letters  a,  b,  c,  d. 

14.  Prove  that  (^a-^b-\-c)(J)-\-c—d){c-^a---b^{a-{-b----c) 
=  2^^V2-:Sa4^ 

15.  How  many  distinct  terms  in  the  result  of  the  last 
example  ? 

16.  Prove  :2'^^.:^^^-7I^^:S'^  =  ^  +  ^+^:. 

0  o^ 

17.  Prove  ^(^-^)-^==3i7(^-^). 

18.  Find  the  value  01  :S'[3— (<:— a)]^  when  «=1,  ^=3, 
and  ^=5. 

19.  Prove  that  {x—d){x—b~){x—c)  .  .  .  equals 

x^—^ax^'-^-V^abx''-'^  —  ,  .  .  +  (—1)"^^^.  .  . 


CHAPTER  V. 

DIVISION. 

93.  Division  is  the  process  of  undoing  multiplication  ; 
that  is,  division  is  the  process  of  finding  from  two  given 
numbers,  called  the  Dividend  and  Divisor,  a  third  num- 
ber, called  the  Quotient,  such  that 

Divisor  X  Quotient =I>ividend,  [1] 

Thus,  to  divide  12  by  3,  we  must  determine  the  factor 
which,  when  multiplied  by  the  given  factor  3,  will 
produce  12. 

In  an  expression  like  a-^b-^c-^d  we  will  understand 
that  a  is  to  be  divided  by  b,  and  this  result  divided  by  c, 
and  so  on.  Since  division  is  the  inverse  of  multiplication, 
either  of  the  expressions  axb-^b  or  a-T-bY.b  is  equivalent 
to  a, 

94.  Law  of  Signs  in  Division.  We  may  write  the 
following  equations : 

(+a&)^(+5)=+a  (1)^ 

(+a&)^  (-&)=-«  (2) 

(-a&)^(+&)=-a  (3) 

(-a&)^(-6)=+a  (4)^ 

from  [2],  Chapter  IV.  Since  a  and  b  stand  for  any  num- 
bers whatever,  we  conclude  that  when  the  signs  of  the 
dividend  and  divisor  are  alike,  the  sign  of  the  quotient 
is  + ,  and  when  the  signs  of  the  dividend  and  divisor  are 
unlike,  the  sign  of  the  quotient  is  — . 

Whence  the  La^v  of  Signs  in  division:  Like  signs 
give  +  and  unlike  give  — . 


[2] 


DIVISION.  49 

95.  Commutative  Law  of  Division.  We  shall  prove 
that  a-r'b-i-c=a-^c-T-b.     For 

a-^b-^cX  (bXc)  =  a-T-b-T'CXcXb, 
by  associative  and  commutative  laws  of  multiplication, 

=  a~bxb, 
=a. 
Also,  a-i-c-^bx  (bXc)  =  a-^c-7-bxbXc, 

by  associative  law, 
=  a-^cxc, 
=  a. 
Whence,    a-irb-i-cx(bxc)  =  a-^c-i-bx(bxc). 

Therefore,  since  these  products  are  equal  and  the  mul- 
tipliers C^xc)  and  (bxc)  are  alike,  the  multiplicands 
must  be  equal  also ;  that  is, 

a^h^c—a-^c-^h,  [3] 

The  same  principle  can  easily  be  proved  for  more  than 
two  divisors,  so  that  we  may  say: 

In  an  expression  co7itaining  several  divisors  the  order  of 
the  divisors  is  indifferent. 
This  is  called  the  Commutative  Law  of  Division. 

Again,  we  know 

axb-^cxc=axb. 
Also,  a-^cxbxc=a-^cx{bxc), 

by  associative  law  of  multiplication, 
=a-^cx{cxb^, 
by  commutative  law  of  multiplication, 
=zaxb. 
Whence,  axb-^cxc=a-T-cxbxc, 

Therefore,  axh^c=a-^cxh.  [4] 

That  is,  if  a  succession  of  nu^nbers  be  connected  by  the 
signs  X  and  -r-,  the  order  of  the  nu7nbers  may  be  changed^ 
provided  each  move  with  its  proper  sign. 


I 


50  UNIVERSITY   ALGEBRA. 

96.  Associative  Law  of  Division.  We  will  first 
prove  that  a  X  {h-T-c)=a  x  b-^c. 

We  know       ay.{b-^c)Y.c=^ay.\{b-T-c)y.c\, 

by  associative  law  of  multiplication, 
=  ax^; 
but  axb-^cxc—axb] 

Therefore,  a x  {,b-^c)  xc=ax b-r-cxc. 

Whence,  aX{b^c)=aXb-^c.  [5] 

Also,       a-i-(b  Xc)x(bx  c)=a, 
but  a-^b-r-cX(bXc)==a'^b-7-cXbXc, 

=  a-7-bxb-7-cXCy 
by  commutative  law  of  division  and  multiplication, 
=a. 
Therefore,  a-^{bxc)=a-^b-^c.  [6] 

I^ikewise  it  may  be  proved  that 

a-^{b-i-c)=a^bxc.  [7] 

From  [5],  [6],  and  [7]  we  say:  In  a  series  of  numbers 

connected  by  the  signs  X  and  -^  a  parenthesis  preceded  by 

the  sign  -r-  may  be  inserted  or  removed,  provided  the  signs 

X  or  -7-  of  all  numbers  within  the  parenthesis  be  changed. 

97.  Index  Law  of  Division.  We  know  «^-T-a^=a2 
=«^~*,  because  a^  xa^=a^;  also  a^-^a^=a^=a^~^,  be- 
cause a^xa^=a^.     In  general, 

a«^a'-=a«-'',  [8] 

if  n  is  greater  than  r,  because  a**~*'xa*'=a**.  Of  course 
n  and  r  must  stand  for  positive  whole  numbers.  That  is, 
the  quotient  of  any  power  of  a  number  divided  by  a  lesser 
power  of  the  same  number  is  equal  to  that  number  with  an 
exponent  equal  to  the  exponent  of  the  dividend  minus  the 
exponent  of  the  divisor. 

This  is  called  the  Index  Law  of  Division. 


DIVISION.  51 

DIVISION   OF   A   MONOMIAlv    BY   A   MONOMIAL. 

98.  The  four  laws  established  above  enable  us  to  find 
the  quotient  of  any  monomial  divided  by  another  mo- 
nomial.    Thus : 

(1)  Divide  Udchyic. 

Since  the  dividend  equals  the  product  of  the  divisor 
and  quotient,  it  follows  that  the  quotient  of  one  monomial 
by  aiiother  monomial  is  found  by  removing  from  the  divi- 
dend all  the  factors  which  occur  in  the  divisor.  By  the 
laws,  the  factors  4  and  <;  may  be  removed  from  the  divi- 
dend in  any  order,  so  that  the  result  is  Zb, 

(2)  Divide  X^abx^'y^  by  Uxy'^ , 

Using  the  index  law,  etc.,  we  obtain  ^ax'^y'^. 

KXAMPIyE^S. 

Divide 

1.  V^ax^  by  ^a,  6.  Vla'^chy  a'^c.  11.  m^ny*^  by  ny'^ , 

2.  28^/3  by  Ac.  7.  42ac'^  by  ac^,  12.  ^x'^y^  by  ^xy^ . 

3.  81^  V^  by  9^2.  8.  lam'^  by  m.  13.  Sc^xy^  by  2c'^y^ 

4.  Ibam'^  by  \ba.  9.  ba^x^  by  a^.  14.  48m'^n  by  m'^n, 

5.  7Qdx^  by  19d,  10.  Sm^n^  by  n^.  15.  72;t:5j/  i^y  72x^, 

DIVISION   OF  A   POLYNOMIAL  BY   A   MONOMIAL. 

99.  Distributive  L#a^v  of  Division.  The  equation 

{oo+y-j-z+.  .  .)-i-n=x-^n-^y-^n-\-z^n+.  .  .  [9] 

is  true.     For  if  each  side  of  this  equation  be  multiplied 
by  n  the  result  will  be 

x-^y-\-2-\-,  .  .:=x+y-j-2+,  .  . 
which  is  true.     Therefore  [9]  is  true. 

That  is,  the  quotient  of  any  polynomial  by  a  monomial 
eqiials  the  sum  of  all  the  tej'ms  found  by  dividing  each  term 
of  the  polynomial  by  the  given  monomial. 

This  is  called  the  Distributive  Law  of  Division. 


52  UNIVERSITY   ALGEBRA. 

100.  From  the  distributive  and  other  laws  it  follows 
that  a  polynomial  is  divided  by  a  Tuonomial  by  removing 
from  each  term  of  the  polynomial  all  the  factors  which  occur 
hi  the  divisor. 

Thus:  (9^3jr— 15^2^2 _|_i2^^3)^3^^^3^2_5^^_|.4^2 

because  (Sa'^—bax+Ax^)  xSax=9a^x — 15a'^x'^  +  12ax^y 
and  the  quotient  is  found  by  removing  the  factors  3,  a, 
and  X  from  each  term  of  the  dividend. 

If  we  wish  to  divide  16a—9bx-i-Scx'^  by  4:X,  the  factors 
4  and  x  do  not  occur  in  some  of  the  terms  and  conse- 
quently cannot  be  removed  from  them,  so  we  can  merely 
indicate  the  division  in  such  cases,  thus : 

(Wa^9bx-{-Scx')-^4.x=---~  +  ~ 

X  4:  4: 

101.  The  Arrangement  of  the  Work  when  a  poly- 
nomial is  divided  by  a  monomial  is  as  follows : 

(1)  Divide  16^4  +  24^^—20^1:2  by  4jr2. 
Divisor,  4;t:2  \J^6x^24:X^ -20x'-  dividend, 

4:X^-\-   6x  —  5       quotient. 

(2)  Divide  5x'^—7x^y-^4x'^y'^  by  dx'^. 
Divisor,  5x^  I  5x^ —7 x'^y -\- Ax'^y"^  dividend, 

x^—lxy-{-^y'^       quotient. 

EXAMPI^KS. 
Divide 

1.  15«2__9^54.i8^9  by  3^2. 

2.  oa^x^ — Sba'^x'^y^ +20axy'^  by  —5ax. 

3.  12a'^x^y^—2Aax^y^  —  18x^-y-i-6xy  by  —6xy, 

4.  2x^y'^—Sxy^+4x^y—y^hySxy^. 

5.  a^x^ySa'^nx'^y-i-San'^xy'^  —  n^xy^  by  anxy. 


DIVISION.  53 

DIVISION   OF   A   POLYNOMIAL   BY   A   POLYNOMIAL. 

102.  Suppose  we  wish  to  divide  x^  +  5x+6  by  x-\-S, 
The  dividend,  x'^  +  6x+6,  is  the  product  of  the  divisor, 
x-\-S,  and  another  factor  (the  quotient)  which  we  wish 
to  find.  Now  we  can  undo  this  multiplication  if  we  can 
write  x^-\-bx+6  so  that  (:r+3)  will  be  a  factor  in  each  of 
its  terms;  for,  by  the  previous  exercise,  an  expression  is 
divided  by  (^  +  3)  if  we  remove  (;r+3)  as  a  factor  from 
each  term.     But  we  may  say: 

;i:2  4-5;r+6=;t;2  +  3;t:+2;t:+6, 
and  by  using  parentheses,   =;r(:r+3)  +  2(;i:-f  3). 
Then,  by  removing  the  factor  (^+3)  from  each  term,  we 
obtain,  (:r2+5^+6)-T-(;r+3)=.r+2, 

which  is  the  required  quotient. 

As  another  example,  let  it  be  proposed  to  divide 
x^  -{-14:X+A5  by  x-{-9.  Now  this  can  be  done  if  we  can 
write  x^-{-14:X+4:5  so  that  x-{-9  will  be  a  factor  in  each 
of  its  terms;  for  an  expression  is  divided  by  (x-j-d)  if  we 
remove  (x+9)  as  a  factor  from  each  term.    But  we  know 

x'^  +  Ux  -\-4:5=x'^  -i-9x+5x+45, 
and  by  using  parentheses ,   =  x(x  +  9)  +  5  (jt:  +  9) . 
Then,  by  removing  the  factor  (^+9)  from  each  term,  we 
obtain,  (x'^  -j-Ux-\-45)^(x+9)=x-\-5, 

which  is  the  required  quotient. 

It  is  noticed  in  each  of  the  above  cases  that  our  process 
consists  in  breaking  the  given  dividend  into  parts,  so  that 
the  divisor  is  a  factor  of  each  part,  and  then  removing 
that  factor  from  each  part. 

103.  We  may  formally  state  the  above  process  thus: 
One  polynomial  is  divided  by  a  second  polynomial,  if  the 

first  polynomial  be  written  so  that  the  second  polynomial  is 
a  factor  in  each  term  of  the  first,  and  this  factor  removed. 


54  UNIVERSITY   ALGEBRA. 

If  it  is  impossible  to  write  a  polynomial  in  this  manner, 
then  that  polynomial  is  not  exactly  divisible  by  the  pro- 
posed divisor,  and  the  division  can  be  merely  indicated. 
Thus,  (x^'i-6x+2)-i-(x-\-S)  is  worked  as  follows: 

x^  -}-6x-i-2=x^  -\-Sx-hSx-\-2. 
Using  parentheses,  =x(x-i-S')  +  (Sx-\-2'). 

Therefore,  (x''+6x-\-2}-^(x+S)=x+^~ 

KXAMPI.KS. 
Divide 

I.  jt;2+3^+2  byj*;+2.  2.  x'^  +  9x+U  by  :r+7. 

3.  ^2_|_4^__45  by  ;r--5. 

x^-{-4:X—4t5=x^—5x-j-9x—45. 
Using  parentheses,  =x{x—5)-\-9{x—5). 

Removing  the  factor  (;i^— 5)  from  each  term  gives  the  required  quotient: 

(x«+4;c-45)-H(jr— 5)=5C+9. 

4.  ^2^3^-10  by:r-2.  5.  x'^-4:x-21  by  ;i;--7. 

6.  x^—4:X—4:6  by  ^+5. 

X*— 4x— 45=^^+5x— 9x— 45. 
Using  parentheses,  =x{x-\-5)—9{x-^5). 

Removing  the  factor  {x-\-5)  from  each  term  gives  the  required  quotient: 

(x^-4x-45)-^{x-{-5)=x-9. 

7.  ^2-_3^_io  by  ^+2.  8.  x^—4:x—21  by  ;^;+3. 

9.  x^—l^x+Ab  by  ^—5. 

5c«-145c+45=xs-5x-9;c-f45. 
Using  parentheses,  =x(%— 5)— 9(x— 5). 

Removing  the  factor  (;c— 5)  from  each  term  gives  the  required  quotient: 

(x«-14x+45)-J-(x-5)=^-9. 

10.  ^2_7^+io  by  ^—2.         12.  3«2_^19^+20by  a  +  5. 

II.  ^2-5^+4  bya-1.  13.  Aa'^ +  Ua-^^  hy  Aa+2. 

104.  Arrangement  of  "Work.  Since  division  is  the 
inverse  of  multiplication,  we  exhibit  in  connection  with 
each  other  the  arrangement  of  work  in  the  two  operations. 


nvisioN. 


55 


MULTIPLICATION,    OR   THE   DIRECT   OPERATION. 


Multiplicand, 
[  multiplier, 
first  partial  product, 
second  partial  product, 
product, 


X    +     4: 

X  +  9 

x^+  4x 


9:r+36 


x^-hlSx+SQ 


DIVISION,    OR  THE   INVERSE   OPERATION. 

Divisor.  Dividend.  Quotient. 

x+4:  )  .;i;2  +  13;f+36  (  jr-f  9 

J  first  partial  dividend,  x'^+  4x 

first  remainder,  9x-\-S6 

second  partial  dividend,  9.r-f-36 

[second  remainder,  0 

In  the  work  in  division  the  process  is  as  follows :  We 
\  first  arrange  dividend  and  divisor  thus : 

x+4)x'^  +  lSx-\-S6  ( 
f  We  next  divide  x^,  the  first  term  of  the  dividend,  by  x, 
[the  first  term  of  the  divisor,  which  gives  x  as  the  quotient. 
fWe  now  multiply  the  whole  divisor  by  x  and  put  the 
f  product,  x^+4:Xy  under  the  dividend.     We  then  have 
x+A)  x^-\-lSx-i-36  (^x 

x''-\-  4x 

[by  subtraction,  9jt:  +  36 

[  We  next  divide  9x,  the  first  term  of  this  remainder,  by  x^ 
I  the  first  term  of  the  divisor,  which  gives  9  as  the  quotient. 
fWe  now  multiply  the  whole  divisor  by  9  and  put  the 
Iproduct,  9;»;+36,  under  the  last  remainder.  We  then  have 
x+4);ir2  +  13.r+36  (:ir+9 
x'^-\-  4x 

9.^+36 
9.;t:-f36 


I  by  subtraction, 
whence  the  quotient  is  ;r+9. 


0 


56 


UNIVERSITY   ALGEBRA. 


The  student  will  observe  that  the  partial  products 
which,  when  combined,  constitute  the  final  product  in 
multiplication,  occur  in  the  work  in  division  as  ''partial 
dividends,"  which,  when  combined,  equal  the  original 
dividend.  So  the  process  of  division  here  used  consists 
merely  in  breaking  up  the  dividend  into  the  component 
partial  products,  from  each  one  of  which  one  term  of  the 
quotient  is  obtained.  Thus  the  above  work  is  merely  a 
convenient  way  of  breaking  :r2  +  13;t:+36  into  the  two 
expressions,  ("partial  dividends"), 

(:r2+4;»;)  +  (9;i:+36), 
and  when  this  is  written  ^(;f +4)4-9(^4-4)  we  readily 
obtain  x-\-%  for  the  quotient,  by  Art.  102. 

105.  We  give  a  few  more  examples  where  multiplica- 
tion and  division  are  exhibited  together,  so  that  the 
student  may  more  clearly  understand  this  manner  of  un- 
doing multiplication. 


(1)             2a   -11 

(2)        a  4-9 

3^  +  4 

a  -5 

6^2_33^ 

^2-1-9^ 

8^-44 

-5a-45 

3a-ll  )  6^2-25^-44  (  3«+4 

^_|_9  )  ^g_|_4^_45  (  ^_5 

6^2-33^ 

a^^^a 

8^-44 

-5^-45 

8^-44 

-5^-45 

Some  prefer  to  write  both  divisor  and  quotient  to  the  right  of  the 
dividend.     Thus : 

(3)     Dividend,     155C«— 445c432  I  3jc— 4     Divisor. 
15x2— 20x  I  5x— 8     Quotient. 

-24x432 
-24x432 


This  arrangement  saves  space,  and  the  divisor  is  where  it  is  readily 
multiplied  by  each  term  of  the  quotient. 


DIVISION.  57 

106.  It  is  very  important  in  the  division  of  a  polynomial 
by  a  poly7iomial  that  both  dividend  and  divisor  be  arranged 
according  to  the  powers  of  a  commoii  letter.  It  makes  no 
difference  whether  the  arrangement  be  according  to  the 
descending  or  the  ascending  powers  of  a  common  letter, 
but  both  dividend  and  divisor  should  be  arranged  in  the 
same  order.  Any  letter  may  be  selected  for  this  purpose, 
but  the  letter  which  occurs  the  greatest  number  of  times 
in  the  given  dividend  and  divisor  is  naturally  preferred. 

If  some  powers  of  the  selected  letter  do  not  occur  in 
the  dividend,  then  it  is  well  to  leave  a  blank  space  in  the 
work  for  every  such  term.     Thus: 

(1)  Dividea2— ^2bya  +  ^.    (2)  Divide  a^  —  b^  by  a—b, 

a^         — b^  I  a-\-b  a^  — b^  I  a — b 


a^-\-ab           1  a—b 

a^—a^b 

a^j^ab-\-b^ 

-ab-b^ 

a^b 

—ab—b^ 

a^b-ab^ 

ab^-b^ 

ab^-b^ 

KXAMPLKS. 
Divide 

1.  x'^+Sx—AO  by  ^-5.  8.  a^+ia—AB  by  a— 5. 

2.  x'^-{-5x—6  byji;— 1.  g.   a'^—4:a—S2  by  a— 8. 

3.  x'^-\-7x—S0  byj»;-3.  lo.  ^2^7^—78  by  a— 6. 

4.  x'^-\-4x—5  byjt:— 1.  ii.   a^  —  121  by  ^  +  11. 

5.  x'^  —  2x—6S  by  Ji;— 9.  12.  x'^—ax—6a'^  by  x—Sa 

6.  ^2_^7.r— 44  by  ;r— 4.  13.  x'^—da'^  by  x—Sa. 

7.  x^—A  by  jf— 2.  14.  ;i;2— 49>/2  ^y  ;r+7j^^. 

15.  a''-i-6ab  +  9b'^  by  a  +  Sb. 

16.  a'^  —  17ab+72b'^  by  a—db. 

17.  2;»;2— 9^+10  by  2;t:— 5.     19.   6;r2+5;t;-21  by  3;t:+7 

18.  3.r2  +  2;t:— 1  by  3jf~l.       20.   9j»;2_64  by  3jr+8. 


58  UNIVERSITY    ALGEBRA.  * 

21.   a^-i-d^+c^—Sadc  by  a  +  d-hc 
Arrange  according  to  the  descending  powers  of  one  of  the  letters, 
say  a.    It  is  important  to  ^eep  this  arrangement  throughout  the  vvork, 
and  to  give  the  powers  of  ^  preference  over  those  of  c. 

—a^b—a^c  —Sabc 

—a^b  —ab^—  abc 


—a^c-\-ab^—2abc 
—a^c  —  abc- 


ab^—   abc^ac^-\-b^ 

ab^ -\-b^^b^c 

—  abc-\-ac^         —b^c 
f  —  abc  —b^c—bc^ 

•  -_  _____ 

ac^  -\-bc^-\-c^ 

Sometimes  the  use  of  parentheses  simplifies  such  examples  : 

a-\-b-\-c 


a^  —  ^abc-Yb^-\-c^ 

a^-Ya^{b-\-c)  \a^  —  a{b-Yc)-\-{b^ —bc-^c^) 


-a^{l)-\-c)  —  ^abc-^b^-Yc^ 

—a^[b  +  c)-a[b-{-c)^ 

a{b^—bc-\-c^)-\-b^-{-c^ 
a{b^-bc-Yc^)^b^-\-c^ 

22.  2^4— 6^3  +  3^2__3^_^l  l3y  ^2_3^_^i 

23.  2w4-6;;^3  +  3;;^2— 3^+1  by  m'^'-Zm-^l. 

24.  4:j/4-18j/3+22ji/2— 7r+5  by  2jj/— 5. 

25.  45.;*:*  +  18;trS  +  35j»;2+4;r-4  by  ^x^+lx—l. 

26.  a'^b'^  —  b^+a^—a'^b^   by  a^ -^ a'^ b ^ b^ -\- ab''- , 

27.  x^+y^+Sxj/—l   by  x-i-y—1. 

28.  ^2—2^^4- ^2 —^24-2^^-^2  by  a—b+c—d, 

29.  a^  +  b^'-c^—2aH^   by  a^  — ^2__^2^ 

30.  l+x^+x"^   by  x^  +  l—x, 

31.  «5_243  by  «~3. 

32.  1—6^5+5^6  by  l_2;t;4-^2 

33.  ;t:^  — 2<2^;t:^+^^  by  x^—2ax-\-a^, 

34.  3.rS-3  by  |^2  +  i^+|. 


DIVISION.  59 

THK    FUNDAMENTAI,   I^AWS   OF   AI.GKBRA. 

107.  The  fundamental  laws  of  Algebra  are  collected 
below  for  convenient  reference. 

A.       ADDITION    AND    SUBTRACTION. 

108.  Commutative  ILaw.  If  a  succession  of  numbers 
be  connected  by  the  signs  +  and  —  the  order  of  the  numbers 
may  be  changed^  provided  each  number  moves  with  its 
proper  sigii. 

109.  Associative  Law.  If  a  succession  of  numbers  be 
co7tnected  by  the  signs  +  and  — ,  a  parenthesis  preceded  by 
the  sign  +  ^^JF  be  inserted  or  removed,  without  changing 
the  value  of  the  expression,  and  a  parenthesis  preceded  by 
the  sign  —  may  be  inserted  or  removed,  provided  the  signs 
+  a7id  —  of  all  the  numbers  within  the  parenthesis  be 
changed. 

B.       MULTIPLICATION    AND    DIVISION. 

110.  Commutative  Law.  If  a  succession  of  numbers 
be  conyiected  by  the  signs  X  aiid  —  the  order  of  the  numbers 
may  be  changed,  provided  each  number  moves  with  its 
proper  sign. 

111.  Associative  Law.  If  a  succession  of  numbers  be 
connected  by  the  signs  X  and  -r-,  a  parenthesis  preceded  by 
the  sign  X  may  be  inserted  or  removed  without  changing 
the  value  of  the  expression,  and  a  pare7ithesis  preceded  by 
the  sign  -r-  Tnay  be  inserted  or  removed,  provided  the  sig?is 
X  and  -r-  of  all  the  numbers  within  the  parenthesis  be 
cha7iged. 

112.  Distributive  Law.  The  p7'odi(ct,  or  qicotie7it,  of 
the  sum  of  several  numbers  by  a  given  7iumber  equals  the 
sum  of  the  respective  products,  or  quotients,  of  each  sum- 
mand  by  the  give7i  number. 


6o 


UNIVERSITY   ALGEBRA. 


113.  Index  La^vs.  The  product  of  two  powers  of  the 
same  number  is  equal  to  that  number  with  an  exponent 
equal  to  the  sum  of  the  exponents  of  the  two  factors.  The 
quotient  of  any  power  of  a  number  divided  by  a  lesser  power 
of  the  same  number  is  equal  to  that  7iumber  with  an  ex- 
ponent equal  to  the  exponent  of  the  dividend  minus  the 
exponent  of  the  divisor. 

« 

MISCEI.I.ANKOUS   KXERCISKS. 

1.  Show  directly  from  the  meaning  of  the  symbols 
employed  that  a-\-{b'-'C)  —  a-{-b—c, 

a—{b—c)  =  a—b-\-c. 
Simplify  the  following  five  expressions : 

2.  a-Z{b-1\_a-Zb'\-1a^. 

3.  a-(3«  +  ^-[4«-(3/^-^)]  +  3^). 

4.  \la-Kh-a)\-\l\{b-\a)^%{a-\\b-\aY)\ 

1c- 


a     r2c—Sa     (      a— 2c     r^      3a4-^~|\~l 


6.  4(a-f[^-|^])(i[2a-^]  +  2[^-.]). 

7.  From  x{x-\-a—2b)(^x—2a-\-b)  subtract  (^x-\-a){x+b) 
(x-2a~2b). 

8.  {x—y-{-z){x-{-y—z)  —  (x-\-y-\-2)(^x—y—2)—Ay2=} 

9.  Divide  ^a''b^-12a^b+U^ -^la^b'' -{-4.a'^-llab^  by 

10.  Divide  (ac+bdy  —  {ad^bcy  by  (^-^)(^-^). 

11.  Dividea^— 0/2— 3<^2^^2_^^_^^2  4_^4   by  ^2_^^_|_2^2 

12.  Divide   x^{a-\-l) —xy(x—y){a-^  b)—y^ib—V)    by 
x'^—xy-\-y'^. 

13.  Divide  (l  +  a)2(l  +  ^2>)_(i_p^)2(i^^2)  bya-^. 

14.  \2a''x''-2{Zb-4.cXb-c)y''-^abxv']-^[ax^2(b-c)y'] 
equals  what? 


DIVISION.  6l 

Historical  Note.  The  way  in  which  Greek  and  Hindoo 
science  reached  the  Occident  during  the  middle  ages  forms  an  inter- 
esting study.  Greek  and  Hindoo  thought  would  have  been  in  danger 
of  being  lost,  or  of  reaching  the  new  nations  of  Europe  much  later 
than  it  did,  had  it  not  been  for  the  Arabs.  As  original  investigators 
in  mathematics  the  Arabs  did  not  excel,  but  they  were  zealous  in 
acquiring  and  recording  in  their  own  language  those  branches  of 
Greek  and  Hindoo  mathematics  which  they  were  able  to  understand. 
When  the  love  of  science  began  to  grow  in  the  Occident,  they  trans- 
mitted to  the  Europeans  the  intellectual  treasures  of  antiquity  in  their 
possession. 

Among  the  many  terms  of  Arabic  origin  is  the  word  ' '  Algebra. ' ' 
Its  earliest  appearance  is  as  the  first  word  in  the  title  of  a  work  by 
Mohamed  ben  Musa  Hovarezmi,  of  the  9th  century,  entitled  Aldshebr 
Walmiikabala.  "Aldshebr"  has  passed  into  "Algebra."  The  two  title- 
words  mean  "restoration"  and  "reduction."  Thus,  ^jc^  — 2a;=5x+6 
passes  by  "Aldshebr"  into  x;2=5xH-2%-j-6,  and  by  "Walmukabala'' 
into  5c2=:7%+6. 

About  the  beginning  of  the  12th  century,  Arabic  MSS.  began  in 
Europe  to  be  translated  into  Latin.  Thus,  Abilard  of  Bath  translated 
the  works  of  Ben  Musa  Hovarezmi.  This  was  done  also  (along  with 
many  other  mathematical  works)  by  Gerhard  of  Cremona.  In  this 
way  Algebra,  with  its  rules  for  solving  linear  and  quadratic  equations, 
was  transplanted  to  Europe. 

It  is  interesting  to  observe  to  what  great  extent  the  progress  of 

[  Arithmetic  and  Algebra  has  been   dependent  upon   the   apparently 

\  small  matter  of  the  kind  of  notation  adopted.    To  peoples  not  familiar 

\  with  the  Arabic  notation,  arithmetical  calculations  with  large  hum- 

f  bers  were  impossible  or  insufferably  tedious.    In  Algebra,  Diophantus 

and  the  Hindoos  used  symbols  (different  from  ours),  but  to  a  much 

less  extent  than  is  done  now.     The  general  introduction  of  new  sym- 

;  bols  has  usually  been  slow.     Thus,  the  mode  of  designating  powers 

I  by  indices  wab  suggested  by  Oresme  in  the  14th  century,  but  remained 

lunnoticed ;  it  was  brought  forward  again  by  Simon  Stevin  in  Holland 

l(died  1620),  but  was  not  appreciated   until  the  time  of  Wallis  and 

Newton.     The   development   of   the   notion   of   a   general   exponent 

(negative,   fractional,   incommensurable)   first   appears  in  a  work  of 

.  John  Wallis  (published  in  1665)  in  connection  with  the  quadrature  of 

1  curves.     The  practice  of  denoting  general  or  indefinite  quantities  by 

[  letters  of  the  alphabet  was  introduced  by  Francois  Vieta  (died  1603), 

\  while  Thomas  Harriot  (died  1621)  first  used  for  that  purpose  small 


62  UNIVERSITY   ALGEBRA. 

letters  in  place  of  the  capital  letters  of  Vieta.  Harriot  also  suggested 
the  use  of  the  symbols  >  and  <  in  the  sense  now  current.  The 
earliest  book  in  which  +  and  —  are  found  is  the  arithmetic  of  John 
Widman  of  Leipsig,  1489,  but  they  did  not  come  into  general  use 
before  the  time  of  Vieta.  Our  sign  for  equality  was  invented  in  1540 
by  Recorde,  the  author  of  the  first  English  treatise  on  Algebra.  To 
William  Oughtred  we  owe  the  symbols  X  for  multiplication  and  ::  to 
designate  proportion.  Descartes  in  1G37  denoted  multiplication  by  a 
dot.  The  sign  of  division,  -f-,  was  used  by  Pell  in  1630;  brackets 
were  employed  by  Girard  in  1G29 ;  the  sign  v  by  Rudolf  in  1536. 

The  custom  of  using  a  letter  to  denote  either  a  positive  or  negative 
number  did  not  become  familiar  to  mathematicians  until  the  time  of 
Descartes  about  250  years  ago. 

'  'The  establishment  of  the  three  great  laws  of  Algebra  was  left  for 
the  present  century.  The  chief  contributors  thereto  were  Peacock, 
De  Morgan,  D.  F.  Gregory,  Hankel  and  others,  working  professedly 
at  the  philosophy  of  the  first  principles ;  and  Hamilton,  Grassman, 
Peirce  and  their  followers,  who  j:hrew  a  flood  of  light  on  the  subject 
by  conceiving  Algebras  whose  laws  differ  from  those  of  ordinary 
Algebra.  To  these  should  be  added  Argand,  Cauchy,  Gauss  and 
others,  who  developed  the  theory  of  imaginaries  in  ordinary  Algebra." 

To  show  the  appearance  of  mathematical  work  before  the  intro- 
duction of  the  common  symbols  we  give  the  following  expression 
taken  from  Cardan's  works  (1545) : 

Bj  v.  cu.  R  1087.  10  I  »^  R  z/.  cu.  Bs  108  m  10, 
which  is  an  abbreviation  for  '  'Radix  universalis  cubica  radicis  ex  108 
plus  10,   minus  radice  universali  cubica  radicis  ex  108  minus  10." 
Or,  in  modern  symbols, 

|^V108+10-l^Vl08-10 
Here  is  a  sentence  from  Vieta's  work  (1615) :     Et  omnibus  per  E 
cubum  ductis  et  ex  arte  concinnatus, 

E  cubi  quad.  +  Z  solido  2  in  ^  cubum,  acquabitur  B  plani  cubo. 
This  translated  reads:     Multiplying  both  members  ("all")  by -£* 
and  uniting  like  terms, 

E^ 
E^-\r2-^=B^. 


CHAPTER  VI. 

MATHKMATlCAIy   INDUCTION. 

114.  In  the  following  pages  we  shall  occasionally 
[desire  to  use  a  method  of  demonstration  known  under 
^  the  name  of  Mathematical  Induction.  We  shall  there- 
i  fore  illustrate  and  explain  the  method  here,  so  that  it 

will  not  be  necessary  to  stop  on  the  method  itself  when 

occasion  arises  for  its  use. 

We  shall  first  illustrate  the  method  by  an  actual  ex- 
I ample  of  its  use,  and  then  the  description  of  it  will  be 
I  intelligible. 

115.  Suppose  the  statement  made  that  the  sum  of  any 
I  number  of  consecutive  odd  numbers  is  the  square  of  the 
lnumber  of  numbers  which  are  added. 

We  readily  see  that     1  +  3=  4=2^, 
1  +  3  +  5=  9=32, 
1  +  3  +  5  +  7=16=42. 
In  the  first  case  there  are  two  consecutive  odd  numbers 
'  added  together,  and  their  sum  is  2  ^ .     In  the  second  case 
there  are  three  consecutive  odd  numbers  added  together, 
J  and  their  sum  is  3^.     In  the  third  case  there  are  four 
[  consecutive  odd  numbers  added  together,  and  their  sum 
;is  42. 

Now,  although  at  present  we  do  not  know  absolutely 

that  the  statement  we  started  out  with  is  true  beyond  the 

sum  of  /oar  consecutive  odd  numbers,  still  from  what  we 

have  observed  we  strongly  suspect  it  is  true  in  any  case. 

;  "Now  let  us  for  a  moment  assume  that  the  statement  we 

\  are  considering  is  true  when  71  consecutive  odd  numbers 


64  UNIVERSITY   ALGEBRA. 

are  added  together;  then,  since  the  >^th  odd  number  is 
easily  seen  to  be  2n—l,  we  have 

1  +  3  +  5  +  .  .  .  +  (2;^— 1)  =  7^2. 
Add  2n-\-l  to  each  member  of  this  equation  and  we  obtain 

1  +  3  +  5  +  ;.  .  +  (2;^-l)  +  (2^  +  l)  =  ;^2+2;^  +  l. 
But,  as  is  readily  seen  by  actual  multiplication, 

.'.  1  +  3  +  5  +  .  .  .  +  (2;^  +  l)  =  (;^  +  l)2. 

The  left  member  of  this  equation  is  the  sum  of  n-}-l 
consecutive  odd  numbers,  and  therefore  if  the  statement 
we  are  considering  is  true  when  n  consecutive  odd  num- 
bers are  added  together,  it  is  also  true  when  n-\-l  con- 
secutive odd  numbers  are  added  together. 

But  we  know  that  the  statement  is  true  when  four 
consecutive  odd  numbers  are  added  together;  hence, 
from  what  has  just  been  shown,  it  is  true  when  Jive  con- 
secutive odd  numbers  are  added  together.  Being  true 
when  ^ve  numbers  are  added  together,  it  is  also  true 
when  SIX  numbers  are  added  together,  and  so  on.  Hence 
the  statement  is  true  universally. 

116.  The  method  just  used  may  be  divided  into  three 
parts:  in  the  first  part  we  ascertain  by  observation  or 
trial  that  the  statement  we  are  considering  is  true  in 
some  simple  case ;  in  the  second  part  we  prove  that  if  the 
statement  is  true  in  any  one  case  it  is  also  true  in  the- 
next  case ;  in  the  third  part  we  deduce  that  the  statement 
is  true  for  every  case  after  the  one  ascertained  by  obser- 
vation or  trial. 

117.  The  student  may  have  heard  of  the  word  induction  as  used 
in  natural  science,  but  the  use  of  the  word  there  is  quite  different 
from  its  use  here.  In  natural  science  some  law  is  observed  to  hold 
in  a  number  of  instances,  and  from  this  it  is  assumed  to  hold  gener- 
ally.    The   more  cases  in   which   the  law  is  observed   to   hold,   the- 


MATHEMATICAL   INDUCTION.  6$ 

stronger  is  the  belief  in  the  fact  that  it  holds  generally;  but  we  cannot 
be  absolutely  certain  that  the  law  holds  in  any  case  except  those  that 
have  been  examined,  and  we  can  never  arrive  at  the  conclusion  that 
it  is  a  necessary  truth.  In  fact,  induction  as  used  in  natural  science 
never  amounts  to  absolute  demonstration,  but  mathematical  induction 
is  just  as  rigid  as  any  other  process  in  mathematics. 

The  important  difference   between   induction  as  used  in  natural 

science   and   mathematical   induction   consists   in    the   fact    that   in 

natural  science  the  second  part  of  the  process  as  described  in  Art.  116 

is  always  lacking,  and  it  is  this  part  that  enables  us  to  give  rigid 

f  demonstration  to  the  fact  we  are  considering. 

118.  When  some  statement  is  found  by  trial  to  be 
rue  in  several  successive  cases  it  is  very  natural  for 
tudents  to  infer  that  the  same  statement  is  true  univer- 
|sally.     As  this  is  one  of  the  most  common  mistakes  of 
I5  students,  an  illustration  or  two  may  well  be  given  here. 
Consider  the  expression  x^-}-x-\-17. 
If  x=  0,  then  x'^-\-x+n=   17,  a  prime  number. 
If  x=   1,  then  x^  f  ^-f-17=  19,  a  prime  number. 
If  x=  2,  then  x'^+x-\-Vl=  23,  a  prime  number. 
If  x=  3,  then  x'^-{-x+Vl=  29,  a  prime  number. 
If  x=  4,  then  x'^+x+Yl=  37,  a  prime  number. 
If  x=  5,  then  x'^+x+n=  47,  a  prime  number. 
If  x==  6,  then  x^+x+n=  59,  a  prime  number. 
If  x=  7,  then  x'^+x+17=  73,  a  prime  number. 
If  x=  8,  then  x'^+x+Vl=  89,  a  prime  number. 
If  x=^  9,  then  x'^+x+n=101,  a  prime  number. 
If  x=10y  then  x^+x+n=127,  a  prime  number. 
If  the  student  tries  values  of  x  greater  than  10  he  will 
till  find  that  the  value  of  x'^+x+Vl  turns  out  to  be  a 
prime  number,  and  the  conclusion  is  nearly  irresistible 
that  this  expression  must  always  represent  a  prime  num- 
ber for  any  positive  integral  value  of  x.     Just  before  this 

conclusion  is  reached  the  student  should  give  x  the 
5— u.  A. 


66  UNIVERSITY    ALGEBRA. 

value  16,  whicli  makes  the  expression  x'^+x+n=2Sd, 
which  is  the  square  of  17,  and  therefore  not  a  prime 
number. 

A  similar  illustration  may  be  given  with  the  expres- 
sion x'^-\-x-\-41,  which  will  be  found  to  represent  a 
prime  number  for  any  positive  integral  value  of  x  less 
than  40.  Again,  the  expression  2jr^+29  is  a  prime 
number  for  any  positive  integral  value  of  x  less  than  29. 

KXAMPI,:^S. 

Prove  by  mathematical  induction : 

1,  12+22  +  32  +  .  .  .+n''=^\n(n  +  l)(2n  +  V), 

2.  2  +  22+2^  +  .  .  .  +  2"=2(2'^-l). 


CHAPTER  VII. 

FACTORS   AND   MUI^TIPI,:^. 

119.  A  definition  of  factor  has  been  given  (see  Art.  7), 

,  and  we  have  already  learned  that  an  expression  may  have 

\  several  factors.  For  example,  the  different  factors  of  lO^t:^ 

are  2,  5,  10,  x,  2xy  bx,  lOx,  x'^ ,  2x^,  5x^.  Of  these  factors, 

2,5,  and  x  may  be  called  Prime,  because  they  cannot  be 

further  factored. 

The  expression  10;i;^  contains  the  prime  factor  x  twice ^ 
so  all  the  prime  factors  of  IOjt^  are  2,  h,  x,  x\  and  as  any 
expression  equals  the  product  of  all  its  prime  factors,  we 
have,  10:^2  =  2x5^^. 

When  an  expression  is  written  as  the  product  of  all  its 
prime  factors,  it  is  said  to  be  Resolved  into  its  Prime 
Factors. 

Resolve  the  following  eight  expressions  into  their 
prime  factors: 

I.  Z^x'^y^,        3.  38a^V^        5.  ISSr^^^        7.  I^Zluv'^w^. 
7,.  150a^<^2       4.  51;;^V^        6.  ^I^abn,        8.  Ih^x^yz^, 
9.  What  are  the  different  prime  factors  of  15<a^^? 

10.  What  are  all  the  prime  factors  of  15a^? 

11.  Resolve  '^a'^b^c^  into  its  prime  factors. 

12.  Find  three  factors  of  Z^x^y^  which  are  not  prime. 


120.  In  order  to  factor  expressions  with  ease,  some 
familiarity  with  certain  powers  and  products  is  necessary. 
Some  of  these  powers  and  products  have  already  been 
given,  and  others  needed  very  soon  we  give  here  before 
proceeding  further  with  the  subject  of  factors  directly. 


6S  UNIVERSITY    ALGEBRA.    • 

121.  We  have  already  learned  (Art.  72)  that 

««a''==^«+'',  (1) 

where  n  and  r  stand  for  positive  whole  numbers,  but 
where  a  may  be  either  integral  or  fractional  and  either 
positive  or  negative. 

Multiply  both  members  of  equation  (1)  by  a^  and  we 
obtain  a"a''a'=a''-^''a\  (2) 

But  the  second  member  of  equation  (2)  is  the  product  of 
two  powers  of  the  same  letter,  and  hence  this  second 
member  equals  a"'^'"^\     Hence 

a««-^-= «-+-+-.  (3) 

By  similar  reasoning  it  follows  that 

a"a''aW=a"-^''-^'-^\  (4) 

and  as  this  reasoning  may  be  extended  as  far  as  we 
please,  it  follows  that  the  product  of  any  number  of  factors  ^ 
each  of  which  is  a  power  of  the  same  number^  is  equal  to  that 
number  raised  to  a  power  whose  index  is  the  sum  of  the  in- 
dices  of  the  factors. 

122.  It  has  been  stated  that  the  exponents  are  any 
positive  whole  numbers,  and  evidently  there  is  nothing 
in  the  reasoning  to  prevent  these  exponents  all  being  the 
same.  .'.  a''a"-=a^''  ox  (a'')^=a*'*, 

and  oTa^'a^'^a^''  or  {cC')  ^ = a^"*^ 

and  so  on.     Therefore,  evidently, 

[W'Y-W'  [1] 

where  n  and  r  are  any  positive  whole  numbers,  but  where 
a  may  be  either  integral  or  fractional  and  either  positive 
or  negative. 

123.  We  have  found  the  product  of  powers  of  a  num- 
ber. We  may  also  find  a  power  of  the  product  of  several 
numbers  ' 


FACTORS   AND    MULTIPLES.  69 

,'.  (aby=a^b\  .      (1) 

Also,   (abY  —  (!^b){P'b)(ab)==ababab=aaabbb=a''^ b^ . 

.-.  {aby=a^b^,  (2) 

And  in  general, 

{aby={ab)(ab)(ab').,.\.o  n  factors  each  of  which  is  ab, 
=  {aaa,.. to  n  factors) (^<^^... to  n  factors), 
^a^'b*', 
,'.  {aby=a''b'\  (3) 

Again,  multiplying  each  member  of  equation  (3)  by  ^, 
we  obtain  (abyc"=a*'b"c'\ 

But  the  first  member  equals  {abcy, 

{abcy'=a''b»c»  [2] 

and  so  on,  evidently,  for  any  number  of  factors.  Hence, 
Ike  nth  power  of  the  product  of  any  number  of  nuinbers  is 
equal  to  the  product  of  the  71  th  powers  of  those  numbers. 

In  a  similar  manner  we  can  find  the  n  th  power  of  the 
quotient  of  two  numbers. 

a 


fa\r_aaa 
\b)b~bV 


to  n  factors  each  of  which  is  , 

b 


Now  the  second  member  of  this  equation  equals 
aaa..Xo  n  factors 
bbb..Xon  factors 

and  this  evidently  equals  -r- 


[3] 


124.  The  simplest  case  of  factors  of  polynomials  is 
where  the  same  factor  is  seen  to  be  common  to  all  the 
terms  of  the  polynomial.  In  this  case  the  polynomial 
may  be  written  in  a  simpler  form  by  dividing  each  term 
by  this  common  factor,  enclosing  the  quotient  in  a  paren- 
thesis as  a  multiplier. 


70  UNIVERSITY    ALGEBRA. 

KXAMPIvKS. 

I.  factor  5ax^  +  i5ax'^+20ax + 60a. 

Here  5a  is  seen  to  be  a  factor  of  each  term ;  therefore,  removing 
this  factor,  we  have 

2.  a^dc+a'^dd^-i-a'^de.  5.    rs^x'^  +  rst^y'^+rs'^-t'^z^. 

3.  auv^  +  buvx-\-uvw.  6.  mxyz+nx'^y'^s'^+rx^y^z^, 

4.  6^3  +  2^4+4^5^  ^    lOa'^b'^-lbab^-lbab'^c. 

8.  3;;^^;^;^ — hm'^y'^ —^m'^ x'^ —  Ibm'^y'^ , 

9.  33r2:r«  +  55r2^V— 66;r2^2^/3^ 

125.  Sometimes  one  factor  is  common  to  some  oi  the 
terms  of  an  expression,  and  another  factor  is  common  to 
other  terms.  In  this  case  you  can  sometimes,  though  not 
always,  simplify  the  expression  by  taking  out  the  factors 
just  referred  to. 

KXAMPLKS. 

1.  'Bsictor  ax + ay — a2+bx+by—b2:. 

Here  the  first  three  terms  have  a  common  factor  a,  and  the  last 
three  terms  have  a  common  factor  d.  Taking  out  a  from  the  first 
three  terms  and  ^  from  the  last  three  terms,  we  may  write 

ax-\-ay—az-}-dx-{-dy—dz=:a{x-{-y — z)-\-d{x-\-y—z). 
Now  it  is  plain  that  in  this  expression  the  factor  {x-\-y—z)  is  common. 
Hence  we  may  write 

a[x-\'y—z)ArK^-Vy—A  =  {fL-{-b){x-^y~z). 
Therefore,  putting  the  expression  we  started  with  equal  to  this  last, 
we  get  ax-\-ay—az-\-bx-^by—bz^:L{a-\-b){x-\-y—z). 

Factor  each  of  the  following  expressions : 

2.  ax-{-1ay-{-Zaz—Ux—^by—^b2. 

3.  x^y'^—x^y+x'^-\-y'^'—y-\-\. 

4.  abxys'^  +abxyz+abxy+abxz'^  +abxz+abx. 

5.  auv'wx-\-buvwx-\-auvwy+biivwy, 

6.  x'^y—ay—2b'^x''"+2ab'^. 


FACTORS    AND    MULTIPLES. 
KXPRKSSIONS   OF  THK   FORM  x'^—O*-. 


71 


126.  In  the  form  x'^—a'^  the  letters  x  and  a  stand  for 
any  numbers  whatever.  We  have  already  learned  (Art.  83) 
that  the  product  of  the  sum  and  difference  of  two  numbers 
equals  the-  difference  of  the  squares  of  those  numbers. 
Therefore,  when  we  have  an  expression  which  we  recog- 
nize as  being  the  difference  of  the  squares  of  two  numbers 
we  can  readily  factor  the  expression  into  the  product  of 
the  sum  and  difference  of  those  numbers. 

BXAMPI^F^S. 

Find  two  factors  of  each  of  the  following  expressions: 

1.  :r2-4.  5.  9^4-9^^  9.  100-25. 

2.  ^2_4^2y^  5    16;^2__36^2^2^    10.  2500—16. 

3.  ^232_25^^         7.  ^2__4^2^4^         II    121-81jir8. 

4.  4^2^2__^4^  8.  «8-««.  12.   625-625a2j»;^ 

18.   9inx^ —Z^ny'^ . 


13.  tV-^^J^^-tV-^^^. 

15.  xyz'^  —  ^xy'^, 

16.  x'^2^—4.y^2^, 

17.  49;^V— 196;z2)/3. 


19.  ax^—ax. 

20.  UVcV'^  +4:UV^W^. 

21.  24^2^6  _54^2y^ 

22.  bOax'^y—l^ax^y, 


KXPREJSSIONS   OF  THE)   FORM   x'^+ax-\-b. 

127.  In  Art.  85  we  learned  that  the  product  of  two 
binomials  whose  first  terms  are  alike  is  of  the  form 
x'^+ax+b,  where  a  is  the  sum  and  b  the  product  of  the 
secottd  terms  of  the  binomials.  The  a  here  used  stands 
in  place  of  and  means  the  same  as  the  a+b  oi  Art.  ^b, 
and  the  b  here  used  means  the  same  as  the  ab  of  Art.  So. 


72  UNIVERSITY    ALGEBRA. 

128.  We  now  take  up  the  reverse  process  of  returning 
from  the  product  to  the  two  factors  which  were  multiplied 
together  to  produce  it. 

We  can  factor  any  expression  of  the  form  x'^+ax+b  ii 
we  can  find  two  numbers  whose  sum  is  a  and  whose 
product  is  b. 

It  will  assist  some  in  finding  the  two  factors  if  the  student  will 
remember  that  when  the  third  term  of  the  given  expression  is  pos- 
itive the  second  terms  of  the  two  factors  have  like  signs,  and  when 
the  third  term  of  the  given  expression  is  negative  the  second  terms 
of  the  two  factors  have  unlike  or  opposite  signs. 

KXAMPI,:^. 

Find  the  factors  of  each  of  the  following  expressions : 

1.  x'^  +  8x+7. 

The  first  term  of  each  binomial  factor  will  of  course  be  x,  and  we 
are  to  find  the  second  terms  of  the  binomial  factors.  To  do  this  we 
must  find  two  numbers  whose  sum  is  8  and  whose  product  is  7. 
Obviously,  7  and  1  are  the  only  numbers  that  have  8  for  their  sum 
and  7  for  their  product.  Therefore,  we  conclude  that  the  two  factors 
sought  are  ^+7  and  x-^-l. 

2.  x^—2x-~S. 

Here  we  must  find  two  numbers  whose  sum  is  —2  and  whose 
product  is  —3.  Obviously,  —3  and  1  are  the  only  numbers  whose 
sum  is  —2  and  whose  product  is  —3.  Therefore,  the  factors  are 
5C— 3  and  ^-j-1. 

3.  .;t:2+20.;ir+100.      7.  x^+4:X-21,  .11.  .;r2-.48.;i;-100. 

/^.x^+2x—8.  B.  x^-16x+48.  12.  x'^-4:X-21. 

5.  .;i;2-9:r+8.  9.  .^ir^  — 10;r+16.  13.  x'^+4x+4. 

e.x^-Ax—m.  10.  .^2  +  11^+13  14.  .r2  +  23;c-50. 

129.  In  the  above  examples  the  first  term  is  x"^  and 
the  second  term  is  x  multiplied  by  some  number!  Of 
course  other  expressions  than  these  might  be  used  for  the 
first  and  second  terms  respectively,  provided  only  that 


FACTORS    AND    MULTIPLES.  73 

the  first  term  is  the  square  of  some  expression  and  the 
second  term  is  that  same  expression  multiplied  by  some 
number. 

EXAMPIvKS. 


I. 

a'^x'^-^ax-Yl. 

7- 

x''y'^-10xy-2m. 

2. 

x^—'lx^^-Zh. 

8. 

a^-na^  +  10. 

3. 

r^x^-16rx''  +  m. 

9. 

4;ir4 +  20x2:1:2+36. 

4. 

n^a^  +  Sln^-a'^+SO. 

10. 

4;t:2+40:r+36. 

5. 

aH^-12aH^-^ll: 

II. 

a'^r^x^^nar'^x^  +  n, 

6. 

m^  +  nm^+SO, 

12. 

a^x^—5ax'^y—60j/^. 

BXPRKSSIONS  OF  THK   FORM   ax'^+dx+C. 

130.  This  form  differs  from  the  preceding  only  in  x^ 
having  a  coefficient  other  than  unity.  We  might  first 
take  out  the  factor  a  and  write  the  expression  in  the  form 

alx'^-l — x-] — j  and  then  proceed  to  factor  the  expression 

in  the  parenthesis  as  before,  but  it  is  perhaps  better  to 
see  how  an  expression  of  the  form  ax'^+dx+c  may  be 
produced  by  multiplication. 

131.  I^et  us  multiply  nx+r  by  sx+t 

nx+r 

sx+t 

nsx'^+rsx 

+  ntx+rt 


nsx'^  +  (rs+nt)x-\-rt 

This  product,  which  is  of  the  form  ax'^  +  bx+c,  we 
must  now  look  upon  as  the  given  expression,  and  the 
factors  v/hich  were  multiplied  together  to  produce  it  as 
the  required  factors. 

A  careful  examination  of  the  work  given  above  will 
show  how  to  return  from  this  given  expression  to  the 


74  UNIVERSITY   ALGEBRA. 

two  required  factors.  We  notice  that  the  first  term  of 
the  given  expression  is  the  product  of  two  factors  which 
are  the  first  terms  of  the  two  required  factors,  and  the 
last  term  of  the  given  expression  is  also  the  product  of 
two  factors  which  are  the  second  terms  of  the  two  required 
factors.  But  probably  the  first  term  of  the  given  expres- 
sion can  be  factored  in  several  ways,  and  the  same  is 
probably  true  of  the  last  term  of  the  given  expression, 
and  how  do  we  know  which  factors  to  select?  The 
middle  term  of  the  given  expression  must  decide  this, 
for  the  factors  must  be  selected  in  such  a  way  that  the 
sum  of  the  products  obtained  by  multiplying  the  first 
term  of  each  factor  by  the  second  term  of  the  other  factor 
shall  equal  the  middle  term  of  the  given  expression. 

132.  Let  us  exemplify  the  method  to  which  this  dis- 
cussion leads  by  finding  the  factors  of  6;r^ +23^+10. 
The  first  terms  of  the  two  factors  may  be  either  6x  and  x 
or  Sx  and  2x,  and  the  second  terms  of  the  factors  may  be 
either  10  and  1  or  2  and  5.  From  these  we  must  select 
those  which  will  produce  2Sx  for  the  middle  term  of  the 
given  expression,  and  by  trial  these  are  found  to  be  Sx 
and  2x  for  the  first  terms  and  10  and  1  for  the  second 
terms.  Hence  the  required  factors  are  3;t:-fl0  and  2jr-f  1. 
We  may  therefore  write  6^2  +2Sx+ 10=  (3;»;-f  10)(2;r-f  1). 

Again,  let  us  find  the  factors  of  6j«;2-f  16ji:+10. 
Here  the  first  and  last  terms  are  the  same  as  before, 
and  so  we  have  the  same  factors  to  select  from  as  before, 
but  different  factors  must  be  selected  to  produce  16x  for 
the  middle  term.  By  trial  we  find  that  the  factors  may 
be  either  6.r-f  10  and  x+1  or  Sx+5  and  2j^;+2,  so  that 
we  may  write  either 

6^2  +  i6;r  + 10=  (6;i;H- 10)  (^4- 1), 
or  6x''-\-16x+10=(Sx+6X'^^  +  2). 


FACTORS    AND    MULTIPLES.  75 

But  these  two  cases  do  not  conflict,  for  in  the  first  case  it 
we  take  out  2  from  the  first  of  the  two  factors  we  have 

(6;t:+10)(jtr+l)  =  2(3;r+5)(;r+l), 
and  in  the  second  case  if  we  take  out  2  from  the  second 
of  the  two  factors  we  have 

(Sx+5X2x+2)=2(Sx+6Xx+l). 

KXAMPI.KS. 

1.  6x^  +  nx+12,      4.  6;t:2+27:r+12.      7.  5x'^  +  Ux+8, 

2.  6x''  +  18x+12.      5.  6x^-}-7Sx+12.      8.  8x''  +  10x+S. 

3.  6;tr2+22;f+12.       6.  5;r2+22;»;+8.         9.  8x'^  +  Ux+S. 

10.  12x'^  +  Sxj/-{-y'^.         12.  9x^-  +  12xy+4:jy\ 

11.  12x^  +  lSxy+6y.     13.  dx'^+S7xy  +  4:j/'^, 

KXPRKSSIONS   OF  THK   FORM   a^—b^. 

133.  By  trial  we  find  that  a^—b^  can  be  divided  by 
ej— <^  as  follows: 

a—b')  a^—b^  (  ^2_f.^^+^2 


aH-b^ 
a'^b-ab'' 

ab'^-b^ 

ab'^-b^ 

Now  remembering  that  the  quotient  multiplied  by  the 
divisor  equals  the  dividend,  we  learn  from  this  division 
that  a^-b^  =  {a-b){a''+ab^b''); 

and  as  a  and  b  stand  for  any  numbers  whatever,  we  may 
make  the  following  statement: 

The  difference  betweeri  the  cubes  of  any  two  numbers  may 
he  expressed  as  the  product  of  two  factors,  one  of  which  is 
the  differe?ice  between  the  numbers  themselves,  and  the  other 
is  the  square  of  the  first  nujnber  plus  the  product  of  the  two 
numbers  plus  the  square  of  the  second  number. 


3. 

x^—a^y^. 

4. 

x^—n^j/^. 

5. 

aH^-c^d^. 

6. 

n^r^  —  n^r^. 

76  UNIVERSITY   ALGEBRA. 

KXAMPI^KS. 

Factor  each  of  the  following  expressions : 
I.  «6-8. 

Here  we  have  the  difference  between  the  cubes  of  two  numbers. 

«6  — (^s)3  and  8=23.    Hence  we  write  the  factors  {a^—2)(a^-\-2a^-[-^). 

2.  x^—y^,  7.  ^x^—21a^y^.      12.  x^y^—1, 

8.  125-^^3^^^        13.  ^a^x^-21a^x^ 

9.  l—a^xK  14.  ^ix^  —  12bx^y^ 

10.  x^—^x^b^c^.       15.  ^a^  —  Ma^x^y^ 

11.  ^a^x^—^a^x^.    16.  x^y^—u^v^w^ . 

Sometimes  an  expression  assumes  the  form  of  the  dif- 
ference of  two  cubes,  after  a  factor  has  been  removed 
from  each  term,  as  in  the  following  examples  : 

17.  a'^x^—a^. 

If  we  take  out  the  factor  a^  there  will  remain  x^—a^,  of  which  the 
factors  are  x — a  and  x^-\-ax-\-a^.  Hence  the  factors  of  the  given  ex- 
pression are  a^,  x—a,  and  x^-\-ax-\-a^.  Hence  the  given  expression 
equals  a^[x—a){x^-\-ax-\-a^). 

18.  x^z''-—y^2'^,       20.  3^532_3^2^8^   22.  a^b'^y'^z—c^y'^z 

19.  a^b^x'^-c^x'^.   21.  2a2^6^6_2^2     23   ha^y^-4Qb^x^ 

2/t2 

24.  r^s^t'^-21r'^s^t^.         25.  ^a'^x^y^ ^— 

^xpr:^sions  of  thk  form  a^  +  b^, 

134.   By  actual  division  we  find  that  a^  +  b^  can  be.' 
divided  by  a+b  as  follows: 

a+b)  a^-\-b^  {a'^—ab^b'^ 
a^  +  a'^b 
—a'^b-^b^ 
'-aH-ab'^ 


ab'^-^-b^ 
ab'^-^b^ 


FACTORS    AND    MULTIPLES. 


77 


Now  remembering  that  the  quotient  multiplied  by  the 
divisor  equals  the  dividend,  we  learn  from  this  division 
that  «8  +  ^3  =  Ca+^)(a2-a^+^2). 

and  as  a  and  b  may  stand  for  any  numbers  whatever,  we 
may  make  the  following  statement: 

The  sum  of  the  cubes  of  any  two  numbers  may  be  expressed 
as  the  product  of  two  factors,  one  of  which  is  the  sum  of  the 
numbers  themselves,  and  the  other  is  the  square  of  the  first 
number  minus  the  product  of  the  two  numbers  plus  the 
square  of  the  second  number, 

Kxampl:^S. 

Factor  each  of  the  following  expressions : 

I.  ^6+8. 

Here  we  have  the  sum  of  the  cubes  of  two  numbers.  a^^a'^Y 
and  8=2^.     Hence  we  may  write 

2.  x^^-y^,  6.  a^b^  +  '^a^b^,  lo.  ^^x^y^+Vlhz^ 

3.  a^x^-\-b^y^,  7.  %x^y^ -\-1"ly^,  11.  a^ b^ c'^ -\- a^ b^ c^ 

4.  m^-\-x^y^,  '   8.  m^-\-n^.  12.  x^y^2^-\-l, 

5.  m^-{-aH^.  9.  64^9+8;tr9j/6.  13.  x^y^  -{■  21x^2^ . 

14.  ^a^b^x^  +  Sa'^b^y^.       15.  64^^  +  125^3^6^^ 
Sometimes  an  expression  assumes  the  form  of  the  sum 
of  two  cubes,  after  a  factor  has  been  removed  from  each 
term,  as  in  the  following  examples: 

16.  a'^x^-\-a^.  23.  bn'^r^x^y^  +  AQn^x^z^, 

17.  x^y^ 2"^ -^ a^ b^ z'^ .  24.  uv'^x^y^-\-uv'^w^, 

18.  a'^bxy^-^a'^-bx'^z^,        25.  128xy^ +54x^2^ . 


19.  Sax'^y-h24:xy^. 

20.  BAx^  +  iexy^. 

21.  lOa^b+SOabx^y^. 

22.  5a^x^y^+6y^. 


26.  10000;t;2+80^^ 

27.  10b'^(x+yy  +  10a^6^. 

28.  24:a^x+81a^x''. 

29.  lS5ax+5am^x^. 


yS  UNIVERSITY    ALGEBRA. 

EXPRESSIONS    OF  THE   FORM  X^  +  a'^ X*^  +  a^ . 

135.  By  actual  multiplication  we  find  that 

{x'^+a'^y=x^+2a^x'^-^a^. 
From  this  we  see  that  the  expression  we  are  now  con- 
sidering, viz.:  x^ -{•  x'^ a'^  +  a^ ,  is  almost  of  the  form  of  a 
perfect  square,  viz.:  the  square  of  j^;^  4-^^;  and,  indeed, 
if  the  middle  term  were  only  2a'^x'^  instead  of  a'^x'^,  the 
expression  here  considered  would  be  a  perfect  square. 
So  we  make  the  expression  a  perfect  square  by  adding 
a'^x'^)  but  if  we  add  a'^x'^  we  must  subtract  it  in  order  not 
to  change  the  value  of  the  expression.  We  may  then  write 

or  using  a  parenthesis,    ={x^  +  2a'^x'^+a^)—'a'^x'^. 

Now  it  is  easy  to  see  that  we  have  the  difference  of 

two  squares,  which  we  have  already  learned  how  to  factor. 

Thus  we  have 

.j»;4  +  a2;tr2+a4:=(^4_f_2^2^2_^^4)__^2^2^ 

=  (^2  +  ^2)2_^2^2^ 

^=(x'^+a'^+ax')(x^+a'^—ax'), 

EXAMPI.ES. 

Find  the  factors  of  each  of  the  following  expressions : 

I.  x^^-^x'^-^^l. 

To  make  a  perfect  square  of  this  expression  we  must  add  95C*,  and 
if  we  add  ^x^  we  must  of  course  subtract  ^x^  afterward.  Hence  we  get 

We  now  have  the  difference  of  two  squares,  which,  as  we  have 
learned  before,  is  equal  to  the  product  of  the  sum  and  difference  of 
the  two  numbers.     Therefore  we  have 

or  as  is  perhaps  a  more  natural  arrangement  of  the  terms  of  the  two 
factors,  (^»+3;c+9)(x«-3;tr+9). 


FACTORS    AND    MULTIPLES.  79 

2.  ^r^-f  4;t:2-f  16.       4.  x^+x'^+1,  6.  ;t;V^+-^V^+J^'^ 

3.  1  +  ^'+^^  5.  81  +  9a24.^4^      7.  ;rV+-^^JV^+y 

8.  x'^+Ax^y  +  lQy^,  II.  16;t:4  +  16jtr2^4  4-16^8. 

g.  a^x^  +  ^a^x^y  +  Uy"^.   12.  16;t;4  +  36;t:2jj/2  +  8iy. 
10.  16;tr4+4/^2^^^2_|.^4^8^   13   x^^+4:X^y^  +  16yK 

Sometimes  an  expression  assumes  the  form  under  dis- 
cussion after  a  factor  is  removed  from  each  term,  as  in 
the  following  examples : 

14.  Snx^  +  Sa''nx''+Sa^n.     17.  lOx^y^  +  lOx^y^  +  10. 

15.  ax^-i-a^x^-^a^x,  18.  4:8-i-12a'^d'^  +  Sa^d^. 

EXPRESSIONS   OF  THE   FORM   X^'—a*', 

136.  We  are  going  to  prove  that  jr"— a"  is  always 
divisible  by  x—a. 

x''^a*'=x*'—ax''~'^-\-ax''~'^—a*', 

=x"-\x—a)  +a(;»;"-'-"a"-'). 

Now  tf  x''~^—a*'~^  is  divisible  by  x—a,  then  plainly 
x"~^C^—a)  +  a(x''~'^—a''~'^')  is  also  divisible  by  x—a.  But 
this  last  expression  equals  x^—a"",  as  we  have  shown. 
Therefore,  if  x—a  exactly  divides  ;tr''~^— ^"~\  it  will  also 
exactly  divide  x^—a'\ 

But  x—a  will  exactly  divide  x^—a^,  therefore  it  will 
divide  x^ — a^,  and  since  x—a  exactly  divides  x^—a"^  it 
will  exactly  divide  x^—a^,  and  so  on. 

Therefore,  whatever  positive  whole  number  be  repre- 
sented by  n,  x—a  will  exactly  divide  x''—a''. 

137.  We  thus  see  that  x—a  is  one  factor  of  x'^—a'^. 
The  other  factor  of  x''—a''  is  found  by  actually  dividing 
x''—a''  by  x—a. 


80  UNIVERSITY   ALGEBRA. 

x—a  )  x^'—a*'  (  x''^^-\-ax''~^-\-a^xf'~* 
x^—ax*'^'^ 

ax^^^^a*" 
ax^-^—a^xf"^ 

a^x^^—a"" 

From  the  work  tlius  far  performed  we  see  that  the  suc- 
cessive terms  of  the  quotient  proceed  according  to  the 
law  made  visible  by  the  terms  already  written,  /.  ^.,  the 
powers  of  x  diminish  and  the  powers  of  a  increase  in  the 
successive  terms  of  the  quotient* 

Since  the  powers  of  a  increase  in  the  successive  terms 
of  the  quotient,  it  is  plain  that  the  highest  power  of  a 
must  be  the  last  term  of  the  quotient.  Moreover,  the 
highest  power  of  a  that  may  occur  in  the  quotient  is  a''"^ 
because  this  expression  multiplied  by  a  gives  a"",  the 
highest  power  of  a  in  the  dividend.  Therefore  the  last 
term  of  the  quotient  is  a""'^ . 

Again,  the  last  in  which  x  appears  is  the  (^— l)st 
term,  or  next  to  the  last  term.  The  quotient  may  then  be 
written,  x''-^ -^ ax""-^ -\- a^ x""'^ -\- ,  .  .+a'^-'^+«"-\  This 
expression,  therefore,  is  the  second  factor  of  x^—a"^. 
Therefore  we  may  write, 
x''—ar-=-{x—d){xr-^^-ax''-''-\-a^x''-^  +  .  .  .+«"-'.r+a"-') 

138.  We  have  already  found  that  x^—a""  is  always 
divisible  by  x—a.  Now  this  dividend,  x''—a"-,  means 
that  the  n  th  power  of  a  is  to  be  subtracted  from  the  n  th 
power  oi  x\  and  this  divisor,  x—a,  means  that  the  num- 
ber represented  by  a  is  to  be  subtracted  from  the  number 
represented  by  x,  and  in  both  dividend  and  divisor  x  and 
a  may  stand  for  either  positive  or  negative  numbers.  We 
may  then  write  any  numbers  or  letters  we  like  in  place 
of  .^  or  a  or  both. 


FACTORS   AND    MULTIPLES.  8 1 

Suppose,  then,  we  take  the  case  where  n  stands  for  an 
even  number,  and  write  — a  in  place  of  a  in  both  divi- 
dend and  divisor.  Now,  because  n  is  an  even  number, 
(—«)''=«''  by  the  rule  of  signs  in  multiplication,  and  if 
this  expression,  {,—ay\  be  subtracted  from  x^  we  get  for 
a  dividend  the  same  expression,  x^—a'\  we  had  some 
time  ago^  and  \i  —a  be  subtracted  from  x  we  get  x-\-a 
for  a  divisor.  Therefore  x-\-a  is  a  divisor,  /.  e.,  2,  factor 
of  x^ — a"^  when  n  is  any  even  nuTnber. 

139.  We  have  so  far  reached  the  following  results: 
x'^—a''  can  always  be  divided  by  x—a,  and  x^'—a*'  can  be 
divided  hy  x+a  when  n  is  any  even  number;  or  stated  in 
another  way,  x—a  is  always  a  factor  oi  x^'—a*',  and  x-\-a 
is  a  factor  of  x''—a''  when  n  is  any  even  number, 

140.  Thus  it  appears  that  the  difference  between  like 
powers  of  two  numbers  can  always  be  factored,  but  it 
must  not  be  supposed  from  what  has  been  said  that  the 
easiest  and  best  way  to  factor  x^'—a"'  is  always  to  take 
out  the  factor  x—a  first,  for  it  may  be  that  the  remaining 
expression  after  this  factor  has  been  removed  will  be  quite 
dif&cult,  while  the  original  expression  will  be  moderately 
easj^  to  factor  if  we  proceed  in  some  other  way.  This  will 
be  fully  seen  in  the  examples  which  follow. 

KXAMPI.KS. 

Factor  as  far  as  you  can  the  following  expressions : 

I.    ^4-^4 

This  we  know  has  a  factor  a—b,  and  also  because  the  exponent  is 
even,  a  factor  a-\-b,  and  if  we  take  out  each  of  these  factors  in  turn 
we  have  left  a^-\-b^.     Therefore, 

a^^b^=,^a-b][a-\-b){a^^b^). 
6~U.  A. 


82  UNIVERSITY   ALGEBRA. 

Or  we  might  proceed  thus:  We  may  regard  a^—b^  as  the  differ- 
ence of  two  squares  and  factor  accordingly.     Therefore, 

the  same  as  before,  as  it  ought  to  be. 


2.  X*-l. 

5.  r*s^-t^.    8. 

1-xV- 

II. 

jr*_j'*  — «*z/*. 

3.  x'--y\ 

6.  16;!;* -81.  9. 

l-x\ 

12. 

16a*-;»:V. 

4.  x*y*—z*. 

«*     .;t* 
w*      16' 

13- 

16Ar*-16j/*. 

14.  a^  —  b^. 
From  the  general  discussion  which    precedes  we  know  that  the 
factors  of  this  are  a—b  and  a^-\-a^b'\-a^b^-\-ab^-\-b^,  and  this  is  the 
only  way  we  have  at  present  to  factor  this  expression. 

15.  x^  —  1.       17.  x^y^—1.      19.  1—x^.        21.  x^y^  —  2^, 

16.  x^—y\     18.  M^v^  —  '^2.    20.  243—32.    22.  S2.;i;5  — ^^ 

23.  Z2x^y^  —M^v^w^ . 

24.  a^.r^jj/^— 322^^z;^ze/^. 

25.  — i— -x-  26, 


jj/^  j,5 


27.  a^—b^. 
This  expression  may  be  regarded  as  the  difference  of  two  cubes, 
and  hence  we  may  write 

Each  of  these  factors  is  of  a  form  already  treated,  and  so  we  know 
the  factors  of  each  factor,  viz. : 

a^^b^—{a—b){a^b), 
and  a*'^a^b^+b^  =  [a^-\-b^-\-ab){a^^b^-ab)', 

therefore,  a^-b^=z{a-b){a^b){a^-]-b^-\-ab)[a^-^b*-ab). 

Or,  if  we  prefer,  we  may  regard  a^—b^  as  the  difference  of  two 
squares,  and  hence  may  write 

Here  again  each  factor  is  of  a  form  already  treated,  and  so  we  know 
the  factors  of  each  factor,  viz. : 

a^—b^  =  {a-b)[a^^ab^b^), 
and  a^^b^=z{a-\-b)[a^-ab^b^y, 

therefore,  a^-b^={a-b)[a^b)[a^+ab-\-b^){a*-ab+b*). 

This  agrees  with  the  result  obtained  before,  as  it  ought  to  do. 


i 


FACTORS    AND    MULTIPLES.  83 

28.  X^  —  1.  30.   --r  —  U^X^.  32.   -^— ^• 

29.  ^^jj/^— ^^.  31.  x^y^2^  —  l,         33.  ;^^Jt;^~r^J/^^^. 

34.  ^7-^7^ 
From  the  general  discussion  which  precedes  we  know  that  the  fac- 
tors of  this  are  a  —  b  and  a^-[-a^b-\-a'^b^'\-a^d^-{-a^b^-]-ab^-\-b^,   and 
these  are  the  only  factors  of  a'^  —b"^  that  we  can  find  now. 
x'^       IC^  XC^  Ij'^ 

35.^V-1-    36.^-^.      37.1-^;t^.    38.  ^V-^' 

39.  a^—b^. 
This   may  be   considered   either  as  the  difference  of  two  fourth 
powers  or  the  difference  of  two  squares.     Taking  it  in  the  latter  way, 
we  may  write,  a^—b^  —  {a^—b^)[a^^b^). 

The  first  of  these  factors  has  been  considered  before,  so  we  know  how 
to  factor  it.     Therefore  we  have, 

—  (^S_^S)(^S_j_^S)(^4_j_^4)^ 

=  {a—b){a-\-b){a^^b^){a^\b% 

/y  8  /fj  O  /v-  o   /« » O  /J J  O 

40.  ^V-1.     41.  j;8-^-  42.  ^8 -^.  43.  x^-uH\ 

44.  a^  —  b'^. 
This  may  be  regarded  as  the  difference  of  two  cubes,  viz. : 

Therefore,  a^ -b^  —  {a^—b^)[a^^a^b^-Yb^\ 

—  {a—b){a^^ab^b^){a^-\-a'^b^-\-b^). 

45,x^y^—j^K     46.^-1.      47.^-^.     4^.u^v^s^i^ 

^^         -^  y^  V^       U^ 

KXPRKSSIONS   OF  THE   FORM   X^'  +  a"". 

141.  We  have  already  found  that  x*''—a**  is  always 
divisible  by  x^a.  Now  this  dividend,  jr"— a'*,  means 
that  the  n  th  power  of  a  is  to  be  subtracted  from  the  n  th 
power  of  X,  and  the  divisor,  x-—a,  means  that  the  num- 
ber represented  by  a  is  to  be  subtracted  from  the  number 
represented  by  x,  and  in  both  dividend  and  divisor  x  and 


84  UNIVERSITY    ALGEBRA. 

a  may  stand  for  either  positive  or  negative  numbers.  We 
may  then  write  any  numbers  or  letters  we  like  in  place  of 
X  or  a  or  both. 

Suppose,  then,  we  take  the  case  where  n  stands  for  an 
odd  number  and  write  —a\n  place  of  a  in  both  divi- 
dend and  divisor.  Now,  because  n  is  an  odd  number, 
(— a)''=— ^",  and  if  the  expression  — a''  be  subtracted 
from  x"  we  get  x''-\-a''  for  a  dividend,  and  if  — a  be  sub- 
tracted from  X  we  get  x-\-a  for  a  divisor.  Therefore, 
x^a  is  a  divisor,  /.  e.,  b.  factor  oi  x'^+a''  when  n  is  any 
odd  ?iufnber. 

142.  We  wish  now  to  see  whether  or  not  x"+a**is 
divisible  by  x—a. 

x''-\-a''=  (^"— ^^)  +  2^'*. 
Now  x^'—a'*  is  exactly  divisible  by  x—a,  and  hence  when 
the  expression  (x*'—a'')-i-'2a''  is  divided  by  x—a  there  is 
a  remainder  of  2a''.  Therefore  (x''—a'')-\-2a''  is  never 
exactly  divisible  by  x—a.  Therefore  x^'  +  a'',  which  is 
the  equal  of  (x''—a'')~\-2a'*,  is  never  exactly  divisible  by 
x—a. 

143.  What  has  been  found  with  respect  to  the  two 
forms  x^'—a*'  and  x''-{-a"  may  now  be  stated  as  follows: 

x^ — a*"  is  always  divisible  by  x — a. 

x^ — a"^  is  divisible  by  x+a  when  n  is  an  even  number^ 
but  not  when  7i  is  an  odd  number, 

x'^-^a^  is  divisible  by  x+a  wheji  n  is  an  odd  number ^ 
but  not  vjhen  n  is  an  even  number. 

x^+a*^  is  never  divisible  by  x — a, 

144.  Although  x"-\-a''  cannot  be  divided  by  either 
x-^a  or  x—a  when  7i  is  even,  yet  it  is  not  stated  that  in 
this   case   x''-\-a''  has   no    factors,   for   sometimes   other 


1 


FACTORS   AND    MULTIPLES.  85 

factors  can  be  found,  as  will  be  seen  by  the  examples  to 
follow,  and  the  explanations  accompanying  them. 

KXAMPLKS. 

Factor  each  of  the  following  expressions: 

I.  a^-Vb^, 

By  the  general  discussion  we  know  that  this  has  a  factor  a-\-b,  and 
another  factor,  found  by  division  oi  a^-\-b^\tj  a-^d,  is 

2.  x^-i-1.  4.  a^x^-i-d^y^.   ,    6.  n^x^-i-S2x^y^z^. 

3.  x,^j/^-{'2^,  5.   1  +  32^^  7.  a^+S2u^v^w^. 

8.  -5+-T.  10.   l  +  oo- 

a^     b^  a^x^     b^z^ 

12.  ^«  +  3«. 

This  may  be  regarded  as  the  sum  of  two  cubes,  and  therefore  we 
may  write,   a^-^b^—{a'^Y^{b^Y  =  {a'^^b^){a^—a^b^^b'^). 

13.  a^-{-b^,       15.  a^z^-^^U^y^.      17.   64:+x^y^z^. 

14.  a^  +  1,         16.  u^v^  +  64:.  18.  x^y^z^+x^^w^. 

19.  -fi  +  1.  20.   -^+-fi. 

21.    ^7+^7/ 

By  the  general  discussion  we  know  that  this  has  a  factor  a-[-b,  and 
the  other  factor  obtained  by  division  will  be  found  to  be 
a^—a^b+a^b^—a^b^-\-a^b^—ab^-\-b^. 

22.  x'^+y\  24.    1  +  x'^y'^z'^,  26.    Zij7^7_|_^7^7 

23.  2^7+1.  25.     128  +  1.  27.     ^7^7^7+128. 

o    2^*^  ,  ;tr7  x'^  ^  y'^ 

28.    -y  +  -Y.  29.    -Y+--7. 

30.  a^  +  b^. 
This  may  be  regarded  as  the  sum  of  two  cubes,  and  hence  we  may 
write,  a^+b^  =  {a^)^-\-{b^)3  =  {a^-\-b^){a^-a^b^-\-b^), 

=  {a-\-b){a^—ab-^b^)(a^—a^b^-{-b^). 


86  UNIVERSITY   ALGEBRA. 

31.  x^  +  1,  33.   X^y+;z^.  35.    u^+v^W^X^, 

32.  ^+^.  34.  a^x^^b^yK      36.  ^+^. 

37.   «io_f_^io^ 
This  may  be  regarded  as  the  sum  of  two  fifth  powers,  and  hence 
we  may  write, 

38.  ^i04._^io^  40.  ^10  +  1.  42.   1024+:^^V^^- 

39.  Ifo+fro-        41.  1024+1.         43.  jro  +  l- 

145.  In  the  expressions  x'^—a''  and  ^''+<2'*  the  ex- 
ponent in  each  term  is  the  same,  but  the  methods  used 
enable  us  in  some  cases  to  find  factors  of  expressions 
in  which  the  exponents  are  not  the  same  in  each  term, 
as  for  instance  in  the  expression  a^-\-b'^^ .  The  exponents 
are  5  and  10  respectively,  but  we  may  here  consider 
^10  =  (^^2)5^  ajj(j  therefore  a^  -\-b'^^  may  be  considered  the 
sum  of  two  fifth  powers,  viz.:  a^  +  (^2-)5^  ^^^  .^^^  ^^^  f^^,. 
tored  the  same  as  in  example  1  of  the  preceding  article, 
using  b'^  the  same  as  b  was  used  before. 

146.  We  add  a  few  miscellaneous  examples  on  expres- 
sions of  the  form  xf—a**  and  x*'-\-a'^,  where  in  some  cases 
a  factor  must  be  removed  from  each  term  before  the 
expression  is  in  either  of  these  two  forms. 

i^XAMPI^KS. 

1.  3^*— 48.  4.  x^+a^y"^^,         7.  Zu^v^—Zu^x^^. 

2.  2«6_32^2^4^      5^  x^—a^y^'^,         8.  hx'^'^-\-hx^y^'^, 

3.  Zx^—Zy^'^,  6.  «i8--^i^  9.  10^8-10<58^8. 

10.  («+^)8~(a2~^2)8^       13^   ^x-Vyy-{x-yy. 

11.  Zmx'^^—Zm^y^  14.   {a'^  +  b'^Y  —  {a'^  —  b'^Y . 

12.  la      ^3.  15.        32  32       • 


FACTORS   AND    MULTIPLES.  8/ 

MISCKI.I.ANKOUS   FACTORS. 

147.  Some  expressions  are  quite  difficult  to  factor 
unless  one  sees  how  the  expression  may  be  changed 
in  form  by  rearranging  or  grouping  the  terms,  or  by 
adding  and  then  subtracting  the  same  expression,  or  by 
some  other  device  to  change  the  expression  into  a  form 
that  will  be  recognized  as  coming  under  some  case 
already  treated.  To  help  the  student  to  see  some  of  the 
devices,  we  take  a  few  of  these  irregular  expressions  and 
work  them  out.  The  student  is  advised  to  go  through 
the  explanation  given  to  see  just  what  is  done  and,  if 
possible,  why  it  is  done,  and  after  two  or  three  cases  to 
cover  up  the  explanation  given  and  see  if  some  method 
suggests  itself;  if  not,  see  how  the  work  is  started  and 
then  try  to  complete  it  without  looking  at  the  rest  of  the 
explanation.  If  still  unable  to  do  the  work,  study  the 
whole  explanation  given. 

(1)  Let  us  factor  a^-^b^-^c^  \-^b^c-^%bc^ . 

We  may  arrange  the  terms  thus:  d^-\-b^-\-^b^c-\-'^bc^-\-c^,  where 
the  last  four  terms  form  a  perfect  cube,  viz. :  the  cube  of  b-\-c.  There- 
fore the  given  expression  may  be  written  a'^-\-{b-\-cY.  We  now  have 
the  sum  of  two  cubes,  and  therefore  the  factors  are  a-\-b-\-c  and 
a^^a[b-\rc)-\-{b-\-cY,  or  «+/^+<;  and  a^-^b^-\-c^—ab—ac-\-'^bc.  Hence 
aZJ^b^J^c^J^U^c^Uc^-{a^b-\-c]{a^^b^-\-c^—ab—ac-\-'^bc). 

(2)  Let  us  factor  a'^-^b^-^c^-\-Za'^b-\-Zab^. 

We  may  arrange  the  terms  thus:  a'^-\-Za'^b-\-^ab^-\-b^-\-c^,  where 
the  first  four  terms  form  a  perfect  cube,  viz. :  the  cube  of  a-\-b.  There- 
fore the  given  expression  may  be  written  [a-\-bY-\-c^.  We  now  have 
the  sum  of  two  cubes,  and  therefore  the  factors  are  a-\-b-\-c  and 
{a^bY  —  {^a-^b)c-^c^,  or  a-Yb-\-c  and  a^-^-b^-^c^-^'lab—ac—bc.  Hence 
a^j^bzj^c^j^^a^b'\-'^ab^-{a-\-b-\-c){a^-\-b^-\-c^-^'lab^ac—bc), 

(3)  Let  us  factor  a^-\-2a^b—a'^—%ab^-^b^—b^. 

This  can  be  arranged  thus:  {a^—b^)  —  {a'^—b^)-\-^ab{a'^—b^),  where 
it  is  plain  that  a^—b^  is  a  factor  of  each  of  the  three  parts  into  which 


88  UNIVERSITY   ALGEBRA. 

the  expression  is  grouped.     Taking  out  this  factor,  the  expression 

may  be  written,  [«2_jaj  [(^2_|_3«)_l_|_2a/5], 

or  [«2__^sj  [i^a^^2ad-\-d»)--l], 

or  [a^-d^][{a+d)^-l]. 

Each  of  these  factors  is  itself  the  difference  of  two  squares,  and  hence 

each  may  be  further  factored  thus: 

Therefore  we  may  write, 

(4)  Let  us  factor  dJ«^2_3*^s^_^8_^s^ 
This  may  be  written  thus: 

or  (^s_^s);,;«_.(^s_^s)^ 

or  («s_^8)(;^s_l) 

or  {a—d){a-\-d){x—l){x-\-l). 

(5)  Let  us  factor  a^-a»-^d^-d^-2a^d^-2ad. 
This  may  be  written  thus: 

{a^-2a^d^+d^)—(a»-{-2ad-{-d^), 

or  (^8_^2)2_(^_|_3)2_ 

This  is  the  difference  of  two  squares,  and  hence  may  be  written  as  the 
product  of  the  sum  into  the  difference  of  the  two  numbers  which  are 

squared.     Hence, 

a*-^2^^4_i,2_2a^d^—2ad 

={a+d){a—d—l){a+d){a—d-]-l). 

(6)  Let  us  factor  a^+d^-^c^—Sadc. 
This  may  be  written  thus: 

or  a^^{6-\-c)^—Sdc{a+d+c). 

The  first  two  terms  are  now  the  sum  of  two  cubes,  and  hence  contain 

the  factor  a-\-d-\-c,  and  this  is  also  a  factor  of  the  last  term,  as  is  very 

evident.     We  may  therefore  write  the  expression  considered  in  the 

form  («+^+V)  [a^—a{d+c)-\-{d+c)^]—3dc{a+d-{-c), 

or  (a-\-d-\-c){a^—ad—ac-{-d^—dc-\-c^). 

(7)  Let  us  factor  a^^b^-{-c^-[-ab^-\-ac^-\-a^b-\-a^c-\-bc^^0^c. 
This  may  be  written  thus: 

[a^J^a^b-\-a^c)-\-[b^-\-ab^+b^c)+{c^+ac^-\-bc^). 


FACTORS   AND    MULTIPLES.  89 

In  the  first  group  a^  is  a  factor  of  each  terra,  in  the  second  group  b"^ 
is  a  factor  of  each  term,  and  in  the  third  group  c"^  is  a  factor  of  each 
term.     Hence  we  may  write, 

And  now  evidently  a-\-b-\-c  is  a  common  factor  throughout.  Hence 
the  original  expression  equals 

(^2_|_32_|_^2)(^_|_^_|_^)^ 

BXAMPI^KS. 

2.  5jr2  +  5jt:— 10.      4.  500j»;2j/— 20y3.    ^.  x^-^^y^ +x-\-1y 

7.  a''b-Vab-a''—a-1b-^2.    9.  hx'^-Xhx^ -"^^x'K 

8.  {c-^dy  +  {c—dy,  10.  l—a'^x'^  —  b'^y'^^-labxy. 

11.  (a  +  ^  +  ^)2_(^_^__^)2 

12.  ^2_|_^2_(^2+^2)_2(j|/<^—^;i:). 

H.  C.  F.  OF   EXPRESSIONS   EASIIvY   FACTORED. 

148.  When  the  same  factor  belongs  to  two  or  more 
expressions  it  is  called  a  Common  Factor  of  those 
expressions. 

When  a  prime  factor  is  common  to  two  or  more  ex- 
pressions it  is  called  a  Common  Prime  Factor  of  those 
expressions. 

The  product  of  any  number  of  common  factors  of  two 
or  more  expressions  is  evidently  a  common  factor  of  those 
expressions. 

149.  The  product  of  all  the  common  prime  factors  of 
two  or  more  expressions  is  called  the  Highest  Common 
Factor  of  thOvSe  expressions.  The  abbreviation  H.  C.  F. 
is  frequently  used  to  stand  for  the  highest  common  factor. 

150.  From  the  definition  of  the  H.C.F.  it  follov/s  at 
once  that  the  way  to  find  the  H.C.F.  of  two  or  more 


90  UNIVERSITY   ALGEBRA. 

expressions  is  to  resolve  each  expression  into  its  prime 
factors,  and  take  the  product  of  all  those  which  are  common 
to  all  the  expressions, 

KXAMPI^KS. 

Find  the  H.C.F.  of  the  following  expressions: 

1.  ma^b^c^,     Iba'^bcd,   and  80^2^^^ 

2.  brs^t^,     Ir^sx^,     ^r'^x^,  and  llrstx. 

3.  llaH^c'^d^  blm^a^b^c'^d^,  and  Ura^b^c^d^. 

4.  x'^yz,     xy'^z,     xyz'^ ,  and  x'^y'^z'^, 

5.  2rstuv,     Zr'^stu,     ^rs^tv,  and  br'^s^tw^, 

6.  Tu^v^w'^y     SBmnu'^v^,     ISQr^u^v^w,    and   ITuv^w. 

7.  l^a'^bx^,     SOb'^ny^,     SQabr^x'^,  and  28b^cdy^. 

8.  ISx^y^z^     90x^yz,     54cxy^,     B&Ox^z,  and   S6xz. 

9.  2x'^'-2xy  and  3;t:2-3j/^ 

10.  x'^—y^  and  ^i;^— -jk^. 

11.  x^+y^  and  ^r^— jj/^. 

12.  ji:^+:r^j/  and  x^-^y^. 

13.  12a^^2j/— 4a^-^J^^  and  SOa^ x^y^ —lOa'^ x^y^ , 

14.  8^'^^V-12a2^^3  and  6a^4^+4a^3(;^ 

15.  x'^—2xS  and  :^2  4.;t;_i2. 

16.  2^(a2-^2)  and  4/^(^-^)2. 

17.  3;rs+6jf2— 24:r  and  6x'^—9ox. 

18.  ^2_9^_io  and  ^2+4^4.3, 

19.  2^^ +2,     3j»;2-3,  and  x'^+Sx+2. 

20.  ;tr2— 3;i;4-2,     :t:2-6:r+8,   and  x'^+x—Q. 

21.  ;»;2j/2— ^2^     ;i:2jj/2_3;rjK<s'+2<8'2,  and  x^y'^—yz^. 

22.  ^*— 8;t:2  +  16  and  ;i;3jk^  +  4;i;2^3_|_4^^3^ 

23.  ;»;3__;^;2^^_i^     ;t;4 _^ ^3 __^2 _^^   ^nd   ;»;2+2;t:— 3. 

24.  x^-x''-4:X+4,     x^-2x'^-x+2,   and   ;i:2+;r-6. 


FACTORS    AND    MULTIPLES.  9I 

H.  C.  F.    OF  KXPRKSSIONS   NOT  KASII^Y   FACTORED. 

151.  The  method  used  above  for  finding  the  H.C.P. 
is  not  appropriate  when  the  expressions  given  are  not 
easily  factored.  ,    - 

152.  When  the  given  expressions  are  not  easily  fac- 
tored the  problem  of  finding  the  H.C.F.  can  often  be 
simplified  by  first  taking  out  from  each  of  the  given 
expressions  all  monomial  factors  and  finding  the  H.C.F. 
of  these,  if  there  be  any,  and  then  finding  the  H.C.F.  of 
the  remaining  polynomial  factors  of  the  given  expres- 
sions. After  this  has  been  done  we  must  multiply  the 
H.C.F.  of  the  monomial  factors  by  the  H.C.F.  of  the 
polynomial  factors,  when  we  shall  have  the  whole  of 
the  H.C.F.  of  the  given  expressions. 

153.  The  method  of  finding  the  H.C.F.  of  the  poly- 
nomial factors  just  spoken  of  depends  on  the  following 
principle : 

If  X  and  Y  represent  two  expressions  and  X^=Q^  Y-\-R^ 
then  the  H.CF,  of  X and  Y is  the  same  as  the  H.C.F.  of 
YandR^. 

We  have  X=Q^Y-^R^  (1) 

.-.  by  transposition,       R^=X—Q^  Y.  (2) 

From  (1)  it  is  evident  that  any  common  factor  of  Y 
and  i?i  is  a  factor  of  X.  Therefore,  any  common  factor 
of  Fand  R^  being  a  factor  of  both  ^and  Kis  a  common 
factor  of  X  and  Y.  Similarly  from  (2)  it  follows  that  any 
common  factor  of  J^ and  Y'v^  a  common  factor  of  Kand  R^. 

We  have  then  two  pairs  of  expressions,  J^and  F  being 
one  pair  and  y  and  R^  the  other  pair,  and  we  have  shown 
that  any  common  factor  of  either  pair  is  a  common  factor 
of  the  other  pair.  Therefore  the  H.C.F.  of  either  pair  is 
the  same  as  the  H.C.F.  of  the  other  pair. 


92  UNIVERSITY    ALGEBRA. 

154.  Now  suppose  we  have  two  expressions  which  we 
represent  by  X  and  F,  and  let  both  of  these  be  arranged 
according  to  descending  powers  of  some  letter  common 
to  both  expressions,  and  further  suppose  that  the  degree 
of  X  with  respect  to  this  common  letter  is  equal  to  or 
greater  than  the  degree  of  Fwith  respect  to  the  same 
letter.  See  Art.  21.  Now  let  us  divide  X\^y  Fand  let 
Q^  represent  the  quotient  and  R^  the  remainder.     Then 

we  have  -^=.Q^-\.-^ 

or  X=Q^Y-^R^, 

By  the  principle  already  proved  we  know  that  the 
H.C.F.  of  X  and  Y  is  the  same  as  the  H.C.F.  of  Y 
and  R-^. 

Now  let  us  divide  Y  hj  R^  and  let  Q^^  represent  the 
quotient  and  R^  the  remainder.     Then  we  have 

R,     ^'^R, 
.-.  Y=Q^R^^-R^, 
and  from  the  principle  alread}^  proved  the  H.C.F.  of  Y 
and  R^  is'the  same  as  the  H.C.F.  or  R^  and  R^y. 

Continuing  this  process,  each  time  dividing  the  last 
divisor  by  the  last  remainder,  we  have  a  series  of  pairs 
of  expressions  which  we  may  represent  as  follows : 

and  by  the  principle  already  proved  the  H.C.F.  of  any 
pair  is  the  same  as  the  H.C.F.  of  the  next  pair;  there- 
the  H.C.F.  of  the  first  pair  is  the  same  as  the  H.C.F.  of 
the  second  pair,  therefore  the  same  as  the  H.C.F.  of  the 
third  pair,  and  so  on.  Therefore  the  H.C.F.  of  the  first 
pair  is  the  same  "as  the  H.C.F.  of  the  last  pair. 


-)    Y  \R,\R,\        R„_,  \ 


FACTORS    AND    MULTIPLES.  93 

155.  Now,  as  this  process  of  dividing  the  last  divisor 
by  the  last  remainder  goes  on,  we  obtain  expressions  of 
lower  and  lower  degree  with  respect  to  the  common 
letter  according  to  powers  of  which  the  expressions  are 
arranged.  Hence  we  must  finally  arrive  at  a  time  when 
the  division  is  exact  or  else  when  the  remainder  does  not 
contain  the  letter.  In  the  first  of  these  two  cases  the  last 
divisor,  being  the  H.C.F.  of  itself  and  its  corresponding 
dividend,  is  the  H.C.F.  of  the  last  pair  and  therefore  of 
the  first  pair  of  expressions.  In  the  second  case  the  last 
pair  and  therefore  the  first  pair  of  expressions  have  no 
common  factor  containing  the  common  letter. 

156.  We  may  now  state  the  method  of  finding  the 
H.C.F.  of  two  expressions  which  cannot  readily  be 
factored  : 

First  remove  from  both  expressions  all  monomial  factors 
and  the7i  divide  one  expression  by  the  other,  the  divisor  by 
the  remainder^  the  last  divisor  by  the  last  remainder,  and 
so  on  U7itil  there  is  no  rernainder.  The  last  divisor  is  the 
H.G.F.  of  the  two  expressions  obtained  after  the  mono7nial 
factors  were  removed.  This  result  multiplied  by  the  H.C.F. 
of  the  monomial  factors  'removed  gives  the  required  H.  C.F. 
of  the  two  expressions  we  started  with. 

157.  It  must  be  remembered  that  this  process  is  not 
to  be  employed  until  all  the  monomial  factors  are  taken 
out  of  the  given  expressions,  so  that  the  H.C.F.  of  the 
remaining  portions  of  the  given  expressions  contains  no 
monomial  factors.     This  being  done,  we  may,  at  any 

I  stage  of  the  process  just  described,   remove   from  any 
dividend  or  divisor  any  monomial  factor  we  please.   Also 
^e   may   introduce   into   any   dividend   or   divisor   any 
knonomial  factor  we  please. 
I 


94  UNIVERSITY    ALGEBRA. 

Rejecting  a  numerical  factor  often  simplifies  the  work 
by  enabling  us  to  use  smaller  numbers,  and  introducing  a 
numerical  factor  often  enables  us  to  avoid  fractions  at 
some  stage  of  the  process.  These  peculiarities  are  illus- 
trated in  the  two  following  examples : 

(1)  Find  the  H.C.F.  of  x^—x^—A  and  3x3-}-;tr2— 4^—20. 

5c3_;r:2— 4  )  3^3_|_  ;^2__4^_20  (  3 

3;c3_3jc2  _i2 

4  )  4x2— 4;c— _8 

jr2— a:— 2  )  x^—x^         —4  (  x  x—2  )  x^—  x—2  (  ^+1 

x9—x^—2x  x^—2x 

2)2x^  x—2 

x—2  ^—^ 

Therefore  x — 2  is  the  H.C.F.  required. 

(2)  Find  the  H.C.F.  of  3^3—4^2+3^—2  and  2a^—Sa9-{-aZ-{-a—l. 
Here  we  use  the  first  expression  for  a  divisor  and  the  second  for  a 

dividend,  and  evidently  the  first  term,  2a^,  of  the  dividend  is  not 
exactly  divisible  by  3^^^  the  first  term  of  the  divisor,  so  we  multiply 
the  dividend  by  3 ;  and  the  work  may  be  arranged  thus: 


3fl-8_4^2_|_3a_2  )  6^4—9^3 


6^4_8^3  I  6^2 — ia 


-3«2+3«— 3  (  2a 


__  ^3_3^2_|_7^_3 

In  a  case  like  this,  where  the  first  term  of  the  remainder  has  a 
minus  sign,  we  change  all  the  signs  before  using  it  as  a  divisor.  This 
is  equivalent  to  multiplying  by — 1.  Making  the  change,  we  continue 
as  follows: 

^3+3^2_7^_^3  )  3^8_  4^2_}_  sa—  2  (  3 
3g3+  9^2—21^+  9 
— 13«2+24«— 11 
Here  again,  the  first  term  of  the  remainder  having  a  minus  sign, 
we  change  all  the  signs  of  the  remainder  before  using  it  as  a  divisor. 
But  even  then  the  first  term  of  the  expression  we  are  to  use  as  a 
divisor  not  being  divisible  by  ISa^,  we  multiply  the  dividend  (which 
was  the  divisor  in  the  operation  just  performed)  by  13,  so  that  the 
division  will  be  exact,  and  continue  as  follows: 


FACTORS    AND    MULTIPLES. 


95 


13 


7«+  3 


13^2—24^+11 


13«3_^39«2—  91^+39  ( a+4 
13^3—24^^+  11^ 


63«2_io2^- 


52^2 


96^+44 


-39 


11«2—     Qa—  5 
Again,  when  we  use  this  remainder  for  divisor  and  this  divisor  for 
dividend,  the  first  term  of  the  new  dividend  is  not  exactly  divisible 
by  the  first  term  of  the  new  divisor,  so  we  multiply  again  by  such  a 
number  that  the  division  will  be  exact  and  proceed  as  follows: 

13^2—  24^+  11 
11 


lla2—Qa—5  )  143^2__264«+121  (  13 
143^2—  78^—  65 

— 186^+186 

Before  using  this  remainder  as  a  divisor  we  will  take  out  the  factor 

-186,  leaving  a — 1,  and  then  proceed  as  follows: 


-1  )  11^2- 
11^2- 


-  6^—5  (  lla+5 
-lU 


5a— 5 

5a — 5 

Since  this  division  is  exact,  we  conclude  that  a — 1  is  the  H.C.F.  of 
the  expressions  we  started  with,  This  is  an  unusually  hard  example, 
and  the  student  who  can  follow  this  will  not  find  any  trouble  with 
any  example  given. 

:examplks. 

Find  the  H.C.F.  of  the  following  expressions: 

1.  Am^—Sm-j-l  and  Sm^+m—l. 

2.  2a^+a'^-^7a—6  and  6a^ -ha^  —  lOa+S, 

3.  8«« -8^2 -4^-3  and  2a^+Sa^-Sa^-^7a-3. 

4.  x^—x'^—x—1  and  2x^+x^-'2x-i-l, 

5.  2;i:s+;r2 +2^-12  and  2x^—7x^  +  Ux-12, 

6.  a^  +  Q7a''+m  and  a^  +  2a^  +  2a^+2a+l. 

7.  28^2+37^-21  and  35a2+62a-33. 

8.  7^«-13w2+34;;^-72  and  Im^-Gm^+SSm—Se. 


96  .  UNIVERSITY   ALGEBRA. 

10.  2x^  +  6x^j/-\-2xy-—y^  and  Sx^  +2x^y-\-xy'^  +2y^ , 

11.  2x^—6x^—2x+6  and  Sx^—dx'^+6, 

12.  x^—x^—7x^+x-\-6  and  ^4+^^— T^tr^— ^+6. 

13.  3;i:3  + 4:^2-13^— 14  and  Gx^+S^r^  — 18;t:+7 

14.  ;r^— Jtr^— ^-— 1  and  x^—x^—x'^  +  l. 

15.  10;i:S+25a^2_5^3  and  4;i:3  +  9^-^^— 2«2^— a^. 

158.  When  we  wish  to  find  the  H.C.F.  of  more  than 
two  expressions  we  first  find  the  H.C.F.  of  any  two  of 
them,  and  then  find  the  H.C.F.  of  this  result  and  the 
third  expression,  and  so  on  until  all  the  expressions  are 
used.  The  final  result  is  the  H.C.F.  of  all  the  given 
expressions. 

BXAMPLKS. 

Find  the  H.C.F.  of  the  following  expressions: 

1.  2x''  +  Sx-5,     Sx'^-x-2,  and   2^2_|.^_3^ 

2.  a^—Sa-2,     2a^-\-Sa'^  —  l,  and   ^^^l. 

3.  12(«4_^4)^     10(a^-d^),  and   SCa^d-ad^). 

4.  x'^—x—12,     x'^  —  6x-\-8,  and  x^—Ax'^—x-\-4:. 

5.  x'^—Qx'^  —  llx—Q,     x^-^4:x'^-\-x—6,     jr^— 3jr+2 

6.  x^—7x^  +  Ux—S,     x'^—6x^+5x+12,    x'^'-5x+4:, 

7.  x^  +  2x'^—x—2,     x^+x^—x—1,  and  x^—x^. 

8.  x^+xy^,     x^y+y"^,   and  x'^-\-x'^y'^  ^y"^. 

9.  ;i:3  +  3.r2~;r-3,     x^—x^,  and  x'^—Zx-^2, 

10.  ;r4— 4;»;2^3,     x^-x^-\-x'^  —  l,     x^'-2x^—x+2, 

11.  x^-\-2x'^'-x—2,     .r^+jr*,  and  x'^+4x-hS. 

12.  ;r2  — 2:^—3.     :i:2-7;r+12,   and  x^-j-x'^-^dx—d. 

13.  x^-^Sx'^'-x—S,     x^  —  2x'^—x  +  2,  and  jr^— Jt:*. 

14.  x^-^-bx'^—x—b,  x^—x'^,   and   ;t:^— ;i:^+;r— 1. 


Mi 


FACTORS    AND    MULTIPLES.  9/ 

LOWEST   COMMON   MULTIPLE. 

159.  Any  expression  is  called  a  Multiple  of  any  one 
of  its  factors.  Thus,  10  is  a  multiple  of  5  because  5  is  a 
factor  of  10 ;  50  is  another  multiple  of  5  because  5  is  a 
factor  of  50 ;  100  is  still  another  multiple  of  5.  10  is 
a  multiple  of  2  as  well  as  of  5,  50  is  a  multiple  of  25  as 
well  as  of  5,  etc. 

160.  A  Common  Multiple  of  two  or  more  given 
expressions  is  any  expression  which  is  a  multiple  of  each 
of  the  given  expressions. 

Evidently,  two  or  more  given  expressions  may  have 
more  than  one  common  multiple.  Indeed,  if  any  common 
multiple  ef  two  or  more  given  expressions  be  found  and 
this  common  multiple  be  multiplied  by  any  expression, 
the  product  will  be  a  common  multiple  of  the  given  ex- 
pressions. 

161.  Any  common  multiple  of  two  or  more  given  ex- 
pressions contains  all  the  prime  factors  of  each  of  the 
given  expressions.  That  common  multiple  of  two  or 
more  expressions  which  contains  the  least  number  of 
prime  factors  is  called  the  Lowest  Common  Multiple 
of  the  given  expressions.  The  abbreviation  L.  C.  M.  is 
frequently  used  to  stand  for  the  lowest  common  multiple. 

162.  From  what  has  already  been  given  it  follows 
that  to  find  the  ly.C.M.  of  two  or  more  expressions  we 
proceed  as  follows: 

Resolve  each  expression  into  its  prime  factors  and  form  a 
product  in  which  each  of  these  prime  factors  occurs  as  many 
.  tim,es  as  it  occurs  in  that  one  of  the  given  expressions  in 
\which  it  occurs  the  greatest  nurnber  of  times. 


98  UNIVERSITY    ALGEBRA. 

Find  the  ly.C.M.  ofSa^x'^y  and  SOax'^. 

Za^x^y  —  ^aax%y\ 

The  prime  factors  are  2,  3,  5,  a,  x,  y.  The  prime  factor  2  occurs 
once  in  the  second  expression,  hence  it  occurs  once  in  the  L.C.M. 
Similarly,  the  prime  factors  3  and  5  occur  once  in  the  L.C.M.  The 
prime  factor  a  occurs  once  in  the  second  expression,  but  twice  in  the 
first  expression,  hence  it  occurs  twice  in  the  L.C.M.  Similarly, 
the  prime  factor  x  occurs  twice  in  the  L.C.M.  Finally,  the  prime 
factor 7  occurs  once  in  the  L.C.M.  Collecting  results,  we  see  that 
the  L.C.M.  is  equal  to  2X3X5rt!^?cxj|/,  or  dOa^x^y. 

KXAMPI.KS. 

Find  the  ly.C.M.  of  the  following  expressions: 

1.  14x2j/2  and  A2xy^2,  4.   ITa'-^^V^  and  17aH^c\ 

2.  7xy^  and  Sax'^y^2;^.  5.   ^Tx'^j/^^^  and  2m'^x^j/. 

3.  9adc  and  15a'^Px'^y.         6.   lix'^j/'^,     9adCj  and  7xj^. 

7.  4:2xy^^,     Sax'^y^z^,  and   dw^. 

8.  a'^ d"^ c'^ x^y^ 2'^ ,   abcic'^v^w^ ^  and  uvwxyzabc. 

9.  hax,    lOay,    lbb'^2'^,    lOOa'^c'^,  and   bOabcxy'^2. 

10.  brsi,    12,    rt,    30,    20r/2,    Ibst^ ,  and   4.sH. 

11.  Uc^d\    21ac\    lOa'^dc,    lOoad^  and    Sobcd'^. 

12.  2bcd\   15ad\   \U^c\   Vdb^d\  21a''c^,  and  35M 

13.  5ac^,    10ac\    7d\    6aH\    15b^c\  and    Uad-^. 

14.  SOaH'^c^  7^ac^d\  A2a''b^d^,  lOhb^c^d^,  S5c^d\ 

15.  20xy'^2  ABxz^v^,  Z^zu^v^  ?>xyzu^v^ ,  and  htc^v^. 

16.  60^z;,   ^hxz'^v,  dOy^zuv,  dxzu'^v,  and  30.rz/^. 

17.  x^  —  1,     x^  —  1,  and  x—1. 

18.  x^—x—6,     x^+x—2,  and  x'^  —  4:X—12. 

19.  4^<^(^2__3^^^2/^2)  and  5^2(^2 4.^^_6^2)^ 

20.  ^— y,     ^+JK,     .^^— j»/^,  and   x^—y^. 

21.  xs— 4.;i:2  4-3;r,     .r*+Jt;'^— 12.r2,     .;r5  +  3x4— 4.;i:8. 


FACTORS    AND    MU-LTIPLES.  99 

22.  x^—7x-}-6,     x'^—5x—6,  and  x^  —  1. 

23.  ;r2 +7^4-6,     x'^-h6x—7,  and  x^— 6jt:— 7. 

24.  5(^2-2^^),     10(^^  +  2^2)^   and    lo^aH^-Ad^), 

I,.  C.  M.   OF  EXPRESSIONS   NOT   EASILY   FACTORED. 

163.  When  we  wish  to  find  the  ly.C.M.  of  two  ex- 
pressions not  easily  factored,  we  first  find  the  H.C.F.  of 
the  two  expressions  by  one  of  the  methods  alread}^  given. 
This  H.C.F.  is  of  course  07ie  factor  of  each  of  the  two 
given  expressions,  and  the  other  factor  is  obtained  by 
dividing  each  expression  in  turn  by  the  H.C.F. 

Now  represent  the  two  expressions  by  X  and  V,  their 
H.C.F.  by  /%  and  their  L.C.M.  by  Af,  and  suppose 

X=Fu  and  V=^Fv. 
Now,  since  /^  is  the  H.C.F.  of  X  and  y,  u  and  v  contain 
no  common  factor.     Therefore 

M=Fuv. 
This  equation  may  be  written  in  either  of  the  forms : 

M=Fuy  =  Xy 
M=-Fv-p  =  Yy 

-.    FuFv    XY 

Therefore,  the  L.C.M.  of  two  expressions  is  found  by 
dividi7ig  either  of  the  expressions  by  their  H.C.F.  and 
multiplyiyig  the  quotieiit  by  the  other  expression,  or  the 
L.C.M.  of  two  expressions  is  fou7id  by  dividing  the  product 
of  the  two  expressions  by  their  H.  C.F. 

Again,  from  either  of  the  last  three  equations,  by  mul- 
tiplying both  members  by  F  it  is  evident  that 

MF=XY, 
i.  e.,  the  product  of  the  H.C.F.  and  the  L.C.M.  of  two 
expressions  is  equal  to  the  product  of  the  two  expressio?is . 


lOO  UNIVERSITY   ALGEBRA. 

Let  us  find  the  L.C.M.  of  Qx^—Ux^y+2y^  and  9jt:3— 22x)/2— S^/S. 
^Q.  first  find  the  H.C.F.  of  these  two  expressions  as  follows: 

3 

9x;3— 225cy*— 8;/3  )  ISx^— SSx^jj/  -j.  6>/3  (  2 

18x3 — 44xy2— 16y3 

—11;/  )  — 33x^;t/+44x>/2+22;/3 
3^2  _  ^xy  —  2y^ 

Sx^—4xy—2y^  )  9x^  —22xy^—8y^  (  Sx-j-iy 

dx^—12x'^y—  Qxy^ 

12x^y—lQxyZ—Sy^ 
12x^y—lQxy^—Sy^ 

From  this  work  we  see  that  3^^ — 4^^;^ — 2y^  is  the  H.C.F.  of  the 
two  given  expressions,  and  if  we  divide  each  expression  in  turn  by 
this  H.C.F.  we  may  obtain  the  other  factors  as  follows: 

dx^—4:xy—2y^  )  Qx^—llx^y  +2y^  (  2x—y 

6^3 —  Sx^y — 4^xy^ 

—  3X2;/+I^j^2_|_2y3 

—  Sx-^y-\-4:xy^^2y^ 

The  second  expression  has  already  been  divided  by  the  H.C.F. 
and  the  quotient  found  to  be  3j^^4~4^-     Hence  we  have 

6x^—llx2y-{-2y^  =  {dx2—4xy—2y'^){2x—y), 
and  9x^—22xyZ—Sy^  =  {3x^—4:Xy—2y^){3x+4y). 

Now  we  have  the  two  given  expressions  factored,  and  from  these 
factors  we  can  readily  write  down  tne  L.C.M.  of  the  two  given 
expressions.     Plainly,  this  L CM.  is 

(3x2— 4;cy— 2;i/2)(2%— ji/)(3x+4>/). 

KXAMPLKS. 

Find  the  ly.C.M.  of  the  following  expressions: 

1.  x^-\-%x'^-\-\^x^-Vl  and  x^ +1x'^ ^^x—Xh, 

2.  x^-V^x'^^X^x+Vl  and  jt^  +  S.^^— 4;r-12. 

3.  x^'\r"lx'^\-nx—\h  and  x^ -\-Zx'^—4.x—Vl. 

4.  bx''  +  llx-j-2  and  15.;r4 +48.^3+9.^2. 

5.  4:X^-10x^+4x+2  and  Sx^-2x^-Sx+2.    ' 

6.  6x'^-\-llxj/  +  4:j/'^  and  4:x'^  —  8xj/'—5y^. 

7.  x^+x'^—Bx+S  and  x^  —  Sx'^-j-Sx-l. 


i(^y 


^       FACTORS   AND    MULTIPLES. 


lOI 


8.  x^+xy'^-\-2y^  and  x^ -\-x'^y-{-4y^. 

9.  x^  +  2x^  +x^  +  Sx'^ ■i-lQx+^  and  x^—4:X^+x'^—4c. 
10.  ;i;5_4^3  4.^2_4  ^nd  x^-^r'^x'^  —  ^x—ll. 

164.  If  we  wish  the  ly.C.M.  of  more  than  two  ex- 
pressions, we  first  find  the  ly.C.M.  of  any  two  of  them, 
and  then  the  L.C.M.  of  this  result  and  the  third  expres- 
sion, and  so  on  until  all  the  expressions  are  used.  The 
result  is  the  ly.C.M.  of  all  the  given  expressions. 

Exampi,e;s. 

Find  the  ly.C.M.  of  the  following  expressions: 

1.  2x'^-\-2x—l,     Sx^—Ax-i-l,   and   2;»;3-3;i:+l. 

2.  ^3+2:1:24-9,     x3_8^+3^   and  x'^-Sx+l. 

3.  x^—2x'-2,     x^—Ax^+S,  and  x^-'Sx'^+2, 

4.  dx^+2x-i-l,     Sx^—8x'^  +  l,  and  x^Sx+l. 

5.  x^-Sx+2,     x^-ex'^  +  llx—Q,  and  x'^-dx+Q. 


CHAPTER  VIII. 

,       FRACTIONS. 

165.  We  have  already  used  the  fractional  form  7  as 

another  way  of  writing  a-r-b.so  that  7  is  an  expression  of 

division.  We  have  already  learned  that  in  any  case  of 
division  the  quotient  multiplied  by  the  divisor  equals 
the  dividend,  or,  in  the  language  of  fractions,  precisely 
the  same  thing  may  be  written, 

Quotient  X  Denomznator=  Numerator. 

166.  From  this  equation  it  is  plain  to  see  that  if  the 
denominator  remains  unchanged,  multiplying  the  numer- 
ator by  any  number  multiplies  the  quotient  by  the  same 
number,  and  dividing  the  numerator  by  any  number 
divides  the  quotient  by  the  same  number,  or,  as  it  is 
more  often  stated,  tnultiplying  the  numerator  by  any  num- 
ber multiplies  the  fraction  by  that  number,  and  dividing 
the  nuTnerator  by  any  number  divides  the  fraction  by  thai 
number. 

167.  Again,  from  the  same  equation, 

Quotient  X  Denominator^  Nurnerator^ 
it  is  also  plain  that  if  the  numerator  remains  unchanged, 
multiplying  the  denominator  by  any  number  divides  the 
quotient  by  that  number,  and  dividing  the  denominator 
by  any  number  multiplies  the  quotient  by  that  number, 
or,  as  it  is  more  often  stated,  multiplying  the  denominato? 
by  any  number  divides  the  fraction  by  that  number,  and 
dividing  the  denominator  by  any  niimber  multiplies  the 
fraction  by  that  number. 


FRACTIONS.  103 

168.  Once  more,  from  the  same  equation, 

Quotient  X  De?iominator^=  Niimerator, 
it  is  plain  that  if  the  quotient  remains  unchanged,  multi- 
plying the  denominator  by  any  number  multiplies  the 
numerator  by  the  same  number,  or,  stated  in  another 
way,  viultiplying  both  7iumerator  and  denominator  by  the 
same  7i2imber  does  not  alter  the  value  of  the  fraction.  It  is 
also  evident  from  the  same  equation  that  dividing  both 
numerator  and  denominator  by  the  same  number  leaves 
the  quotient  unchanged,  or,  stated  in  the  usual  form, 
dividing  both  numerator  a?id  denominator  by  the  same 
number  does  not  alter  the  value  of  the  fraction. 

169.  When  the  numerator  and  denominator  of  a  frac- 
tion have  no  common  factor  the  fraction  is  said  to  be  in 
its  Lowest  Terms.  Therefore,  to  reduce  a  fraction  to  its 
lowest  terms  we  divide  both  numerator  a7id  denomi7iator 
by  their  H.C.F.,  for  by  so  doing  we  obtain  a  fraction  in 
which  the  numerator  and  denominator  contain  no  com- 
mon factor. 

EXAMPLKS. 

Reduce  the  following  fractions  to  their  lowest  terms : 
(3.r-2y)^-(2.y+2j/)^  aM;^^M- 11^  +  6 


I. 


2. 


;t:2-3jr-70  '  30^=^- 19^2  _^1 

{a  +  by-{c-Vdy  ^^-10^2+26^-8 

{a-^cy-{b^dY  ^'  a3~9^2_^ 23^-12* 

3;j;3_e^2_|_^_2  8^^— lO^2__i0^_3 


I     4.  — z^r-n-v^ —  ^o 


x^-lx-\-^  '  6^4 -22^^  + 31^2 _23^_7 

5-       x^-x-Q     '  "•  3^4  +  14^3-9^+2' 

x^-i-a^  1  +  2^-3^2 

6.    -^r-^ —7;'  12. 


x^+2ax+a^  '  1-3^-2^2+4^3 


I04  UNIVERSITY   ALGEBRA 

ADDITION   OF   FRACTIONS. 

170.  If  two  or  more  fractions  have  the  same  denom- 
inator, the  fractions  may  be  added  by  adding  the  numer- 
ators and  placing  the  sum  over  the  common  denominator; 
but  if  the  denominators  are  not  the  same  we  must  mul- 
tiply the  numerator  and  denominator  of  each  fraction  by 
such  a  number  as  will  make  all  the  denominators  the 
same,  and  then  add  the  numerators  and  place  the  sum 
over  the  common  denominator. 

171.  The  process  of  changing  the  numerators  and 
denominators  of  fractions  so  that  each  fraction  shall  pre- 
serve the  same  value  it  had  before  while  the  denomina- 
tors are  all  made  alike  is  called  Reducing  to  a  Common 
Denominator. 

For  example,  suppose  we  wish  to  add  -^  and  — .  We 

iw  mn 

must  reduce  to  a  common  denominator,  which  of  course 
must  be  a  common  multiple  oi  rn^  and  mn.  Any  common 
multiple  will  do,  but  the  lowest  common  multiple  is 
preferable. 

The  lowest  common  multiple  is  plainly  m'^n\  hence 
we  must  multiply  the  numerator  and  denominator  of  the 
first  fraction  by  n,  and  the  numerator  and  denominator 
of  the  second  fraction  by  m.     We  then  have 
abn      4mx 
m'^n     m'^n 
Plainly,  then,  the  sum  of  the  two  fractions  is 
adn  -^Amx 
fn'^n 
Hence  we  may  write  the  equation, 

ad      4:X  _abn  -i-4mx 
ni^     mn  Tn'^n 


FRACTIONS. 


lOS 


If  any  number  of  fractions  are  to  be  added  together  we 
pursue  a  similar  method.  Therefore,  to  add  several  frac- 
tions together,  i^educe  all  Hie  fractions  to  a  conimoji  deiioTti- 
inator,  preferably  the  lowest  commoji  denominator,  add  the 
resulti?tg  numerators  and  place  their  sum,  over  the  common 
denominator. 

KXAMPLKS. 

Add  the  following  fractions : 
a-\-b  J      a-{-b 


2. 
3. 

4 
5. 
6. 


x'^  -j-y'^ 

x^-y^' 

1 


and 


x{x—b^ 
,  and 


x'^—y'^ 


x—y 


2  ,     3 

a-^b-\-c      a  +  b  a-\-c 


a  +  1 


2a  +  S 


and 


3^  +  4 


^2+5a-f6'     (^  +  2)2'  ""^  (a  +  Sy 
2a+6  ,        2^  +  7 


a 


and 


^2+7^  +  10 
b  .      1 


^3_^3>       ^2+^^_^^2' 


and 


a — b 


X 

^*  x+a' 


X  ^        x'^-\-c'^ 

— ,-7,   and  -^- 1 

x+b  x^-\-ax-}-ac 


8. 


x^+a'^ 


x-\-a 


x^-i-a'^x'^  +  a^'     x^-hax+a 


;,  and 


x—a 


x^—ax+a^ 


a'^-bc     b''- 


bc 


ac 


-ac         -   c^ 
and  — 


-ab 


ab 


10. 


^2— 3^+2'    ^2— a— 2 


,  and 


SUBTRACTION   OF  FRACTIONS. 


172.  If  one  fraction  is  to  be  subtracted  from  another 
^and  the  fractions  have  the  same  denominator,  we  may 
[subtract  the  second  numerator  from  the  first  and  place 
I  the  remainder  over  the  common  denominator;  but  if  the 


I06  UNIVERSITY    ALGEBRA. 

denominators  are  different,  we  must  first  reduce  the  frac- 
tions to  a  common  denominator  and  then  perform  the 
subtraction. 

KXAMPI^BS. 

^         2a-\-l  ^  ,     Sa-i-2 

1.  From  — o —  take  — ^— • 

2.  From  ^^  take  ^^. 

a  a 

3.  From  ^±^  take  ^^. 

x—y  x-\-y 

^^-f-1/^  x~\-\ 

4.  From  -::^r^^  take  ^^ 


5.   From  — — -^  take 


0.   From     ^  .  ^ — r^  take 


Jtri/  x"^  V  ~\~  xv^ 

7-  From  ^^  take  -^,^^- 

8.  From  ^!d^Z!  take  ^^%. 

9.  From  £!±^Z!  take  £!z:^2^Z!. 
10.  From  7 ^7 r;  take 


{x—a^ia—b)  (x—b)(a—b) 

II.   From     »  ,    .     ,  ,. — ^ — 7  take 


x'^+  (a-\-b)x-\-ab  x'^ -^(a-^c)x-\-ac 

In  the  following  examples  perform  the  additions  and 
subtractions  indicated  and  express  the  result  as  a  single 
fraction  in  its  lowest  terms : 
1.1  1 


12. 


x+l^x+2    x+Z 


x+1  x+2  x^Z 

^3-    /^  I  o^/<^_LQ^"^/ 


(^+2)(;c  +  3)^(^+l)(^+3)     (;^+l)(^+2) 


fa     b      c\     fa     3^     hc\ 
^5-  I2  +  3-4J+I4-T-6/ 

^    a-^b  ,  b—a        iab 
16. 7+ 


FRACTIONS.   ^  107 

2  3 


17- 


a—b     a-\-b     a'^ — b'^ 

2^+1 3jt;+2 4;r4-3 

(:r-l)(jr-2)     (jt:-2)(jt:-3)     (;r-3)(;t:-4) 


a         <a^4-l         <^        <^^ — a  +  1 

3^—6 4a— 5 a—\ 

2^4-1       3a  +  2 2 ^_3_ 

MUI^TlPIvICATlON   OF   FRACTIONS. 

a        c 
173.  We  are  to  multiply  ;/  by  -• 

Now  to   multiply  by  -7  means  to  multiply  by  c  and 

^  a        c 

divide  the  result  by  d.     Therefore  to  multiply  -r  by  -1  we 

first  multiply  -y  by  <:  and  then  divide  the  result  by  d.    We 

may  multiply  a  fraction  by  multiplying  the  numerator; 
a  ac 

hence,  t><^=^* 

b  b 

We  may  divide  a  fraction  by  multiplying  the  denominator; 

ac      ,     a<: 

hence,  S^'^^M 

^     ^.        r  a      c     ac 

,     Therefore,  t^^— t:?* 

■^  '  b     d     bd 

Hence,  to  multiply  two  fractions  together,  multiply  the 
numerators  together  for  a  new  numerator^  and  the  denom- 
inators together  for  a  new  denominator. 


I08  UNIVERSITY   ALGEBRA. 

174.  Suppose  we  wish  the  product  of  three  fractions, 

-     .     ,  ace 

as  for  instance,  t  x  i  x  t* 

oaf 

We  may  multiply  the  first  two  fractions  together,  as 

ac 
just  explained,  giving  — ,  and  we  can  then  multiply  this 

result  by  the  third  fraction  by  the  method  just  explained, 

.   .       ace      ^.        ^        ace     ace 
givmg^.     Therefore,  ^x^x^=^. 

As  this  process  may  be  extended  to  any  number  of 
fractions,  we  have  :  The  product  of  any  mimher  of  fractions 
is  found  by  multiplying  all  the  numerators  together  for  a 
new  numerator  and  all  the  denominators  together  for  a  7iew 
denominator. 

The  result  should  be  reduced  to  its  lowest  terms  if  it 
is  not  in  its  lowest  terms  already. 

175.  Instead  of  actually  performing  the  multiplications 
it  will  frequently  be  best  to  indicate  them  by  using  paren- 
theses, for  sometimes  in  the  result  the  numerator  and 
denominator  will  contain  a  common  factor,  which  can  be 
struck  out  and  thus  save  the  trouble  of  multiplying  by 
these  factors.     For  example,  if  we  wish  the  product  of 

a+x        b  ^   c — X 

— T-,    —, — ,  and  — -— , 
b         c-j-x  a-\-x 

we  may  write  the  product  thus : 
b{a-\-x){c—x^ 
b{c+  x^{a-\-  x) 
Here  the  numerator  and  denominator  contain  the  com- 
mon  factor   b{a-\-x^,  which   being   rejected   from   both 
numerator  and  denominator  leaves  the  result  simply 

c — X 
c+x 


FRACTIONS. 


109 


KXAMPLKS. 
Find  the  product  of  the  following  fractions : 


4 
5 
6 

7 
8 

9 
10 


1  +  ^2 


a  +  ^2' 
^3 — ^l 


2ab     ' 
a(a  +  2d) 

/,4 


and 


and 
and 


1-a 
1 

^2 


a^— 2^;»;H-;«;2 


^^2 


^34-^3' 


and 


4:a{a  +  b) 


and 


2«2;,;2_|.^4  a2+ji;2 

and 


(3^3;i:+ajr3 


a^— ^^ 


«3  +  3a2^4-3a^2_j.33        a—b 


a'^  +  b'"' 


and 


^2_9jr-f  20     ^2__i3^_|.42 


x^  —  5x 


a 

'a+b 
^2 
and =• 

;i:— 7 


2~_y2 


^-1 


^^,  and  (^-•^>' 


;r2— 2;t:+l 
^^2-9     ' 


and 


x^  -{-y^ 


:^;2-4jr+3 


DIVISION   OF   FRACTIONS. 
(t  C 

176.  We  are  to  divide  -  by  -. 
o       a 

We  may  write  the  quotient  in  the  form  of  a  fraction, 

where   the   numerator  is  itself  the  fraction  -7  and   the 

c  ^ 

denominator  is  the  fraction  ^. 

a 


Hence, 


—=  quotient. 
1 


no  UNIVERSITY    ALGEBRA. 

Let  US  multiply  both  numerator  and  denominator  of 

this  fraction  by  bd.     We  know  this  will  not  change  the 

value  of  the  quotient. 

a      .  ,     abd        . 
Xbd=—r-=ad. 
0  o 

c      ,  ,     bed     , 
-■^bd-=-^-=^bc, 
d  d 

Hence,  quotient=-r— 


.   T;herefore, 


a     c     ad 
b  '  d     be 
This  result  may  be  obtained  by  7nultiplytng  the  fraction 

^by  the  fraction  — ,  which  last  is  the  divisor  inverted. 
b  e 

Hence,  to  divide  one  fraction  by  another,  invert  the 
terms  of  the  divisor  and  multiply. 

EXAMPLES. 
I.   Divide  f!±2^J^^  by  ^-^^+-^^+— -• 

,.  Divide  £!±^^^£!  by  _£-^4_. 
x-\-y — z  x'' — lxy+y''—z^ 


3.  Divide  ^^  by  ^^-^ 

4.  Divide  -^^  by  ^-3- 

5.  Divide  ^;±^  by  ^±^. 

6.  Divide o  ,  o    n — a2 —  ^y  — TTT-' 

a^-i-za-^-l  —  b^  ab+1 

^.    . -,         1       ,         1—a 

7.  Divide  z ^  by  yj-. — :-^- 

1—a^        (l-]-ay 


FRACTIONS.  Ill 

8.   Divide 2 — A2 ^y     r^xh  ' 

a^  —  b^  a-\-o 

^.    .,    a^  —  b^-\-a—b.      a—b 
9-  Divide  ^^-^^^^  by -^. 

,0.  Divide  ^^^  by  :^-±^. 
{x—y}^ — 2^         x—y-\-2 

II.   Divide T — 77 by 


^2-^2 


12.   Divide 1 —  by 


x—l  l-\-ax  ' 

MISCKI^IvANKOUS   FRACTIONS. 

177.  We  may  take  an  integral  expression  (/.  e.,  a 
monomial  or  polynomial  in  which  there  is  no  fraction 
involved)  along  with  one  or  more  fractions,  giving  us  a 
form  partly  integral  and  partly  fractional.  Such  expres- 
sions are  sometimes  called  Mixed  Expressions  or  mixed 
numbers.  Mixed  expressions  may  always  be  reduced  to 
the  form  of  fractions  b}^  writing  the  integral  part  in  the 
form  of  a  fraction  with  a  denominator  1,  and  then  per- 
forming the  indicated  operations.  For  example,  suppose 
we  wish  to  express 

x^—y^ 

I      in  the  form  of  a  fraction.    We  write  the  expression  thus : 

x'^  +y'^         xy 

1  x'^—y'^ 

Reducing  to  a  common  denominator  and  adding,  in  the 

x^—y^+xy 
usual  way,  we  get         ^ — ^  -^• 

This  process  is  called  Reducing  Mixed  Expressions 
to  Fractions. 


112  UNIVERSITY    ALGEBRA. 

KXAMPIvKS. 

Reduce  the  following  mixed  expressions  to  fractions : 

3                           11  1 

2.  x-{-a 4.  x+y-\ 1 —       6.  ax+by 


a  X    y  '  ax— by 

,  r  ,        b     c     a  ,  T  ,      ,         1 

7.  a  +  b+c -r 9.  a  +  b+c-{ r^-r' 

a     b      c  a+b-^c 

S.  x'^+x+l+~'^'  10,  x'^—ax+a^+    ^^ 


x+1  '  x—a 

178.  When  the  given  mixed  expression  has  only  one 

fraction,  as  in  the  example  x^-{-y^+    »       o>  we  reduced 

x^ — y^ 

to  a  common  denominator  by  multiplying  numerator  and 

x^  ~\-  v^ 
denominator  of  — —^  by  jr^  —y'^ ,  the  denominator  of  the 

fractional  part  of  the  given  expression.  This  amounts  to 
multiplying  the  integral  part  by  the  given  denominator 
and  adding  the  numerator,  exactly  as  in  Arithmetic. 

Since  the  fraction  was  produced  from  the  mixed  ex- 
pression by  multiplying  the  denominator  by  the  integral 
part  and  adding  the  numerator,  it  follows  that  to  reverse 
this  process  and  go  back  from  the  fraction  to  the  mixed 
expression  we  would  divide  the  numerator  by  the  denom- 
inator as  far  as  possible  and  write  the  quotient  for  the  inte- 
gral part  and  the  rernainder  over  the  deyiominator  for  the 
fractional  part,  which  again  is  exactly  as  in  Arithmetic. 

KXAMPI.BS. 

Change  the  following  fractions  to  mixed  expressions: 
x^ — x'^y-\-xy'^—y^  x^-\-  Zax'^ -\-  Sa'^x-\-a^-{-a'^ 

x-\-y  '  x-\-a 

x'--\-ax+a'^  x^+a'^x'^-^a^-^x+a 

x'^+a  '  x'^ -\- ax -i- a^ 


FRACTIONS. 


113 


5. 


6. 


10 


r:2  — 1/2 


7. 


■JV     +-2'^ 


ji; 


2— _y2 


8. 


x+y 


x-\-y 


9. 


10. 


4a 


179.  By  combinations  or  repeated  applications  of  the 
preceding  processes  we  are  able  to  deal  with  more  com- 
plicated cases  than  have  yet  been  given. 


For  example,  let  us  take  the  fraction 
x-\-a     x—a 
X — a     x-\-a 


The  numerator 


x-^a     x—a 
x  —  a     x-{-a 
x-\-a     x—a_{x-\-a)^—{x  —  a)* 
x—a     x  +  a  x^—a^ 

_[x^  +  2ax-^a^)-  {x^ -2ax-\-a^) 
~"  x^—a* 

4ax 


The  denominator         = 


x-\-a     x—a 
x  —  a     x-\-a 
_{x-{-a)^-\-{x—a)^ 

x^—a» 
_{x^  +  2ax  +  a^)-\-{x^-2ax+a^) 
x^—a^ 


x^-a^ 
Therefore  the  original  fraction 

_    Ux     ^  2(x^ 


.a^) 


x^—a^ 


iax  x'^—a^ 


A^ax 


2ax 


~2(x^-\-a2)~x»  +  a^ 

This  final  result  is  much  simpler  than  the  fraction 
we  started  with,  so  this  kind  of  work  may  be  called 
Simplifying  Complex  Fractions  or  Expressions  In- 
volving Fractions. 

8— U.  A. 


tI4  UNIVERSITY   ALGEBRA. 

KXAMPI<KS. 
Simplify  the  following  expressions : 


4. 


X  a  +  d 

;»;— 2      ,2x'^-\-X'-'l  ^1,1 

0.  :; 1 — 


a^^b^  a'^+ab+b'^  a'^  +  ab+b'^ 

10.  -^ ^    /  .   70  X 


•  (a-^cy-i^b+dy    {a-cy-^b-dy 
,3  M L.U -ll 

\n—r     n—s)     ?i'^—n(r+s)-\-rs 

^    \   J/  x—y    I     \x    y) 

fa^  —  b^     a^-^b^\^     4ab 
^5-  \-JZrf       a  +  b  )  *  ^2-/^2* 


FRACTIONS.  1 1  5 


/      x^—a^        ^x^+ax\     x^—a^x^  ^  /x     a\ 
\x'^—2ax+a^  '    x—a  J       x^+a^     '  \a     x) 

I+--2  4+4 

18.  ^-^  «.   /'/\ 

a—b  1       1,1 


1  a^     ay    y'^ 

1- 


^-^  1 


X  a        (a—b)^-'4tab 

x^  a      b      a^^  b^     ab 


20. 


{am---bn')'^-\-(an  +  bm)^      23.  — 


a— 1  + 


A-a 
Fractions  like  example  23  are  called  Continued  Fractions. 

To  simplify  a  continued  fraction,  begin  at  the  lowest  part  and  pro- 
ceed upward  step  by  step  as  follows: 

4— «+«       4 


^+4: 

-a 

4— a       4- 

-a 

Hence  the 

original 

fraction 

may 

be  written 
1 

a— 

1-1 

4-« 

But 

4 
4- 

4-a 
_~    4 

a 

Hence  the 

original 

fraction 

may 

be  written 
1 

4-a 


4_^    4^_4_l-4— a    3tf 

But  tf_l  +  _— -= =-— 

4  4  4 

1      4 
Hence  the  original  fraction      =— =— 

T 


Il6  UNIVERSITY   ALGEBRA. 

b 
1  a  ^  b'^ 

24. -. —        25.  7 z —        26 


b  ^       a  '  b^ 


^  + T-Ti  -+ 12  ^+ 


3iy  M-T 

— ^  ^  a 

28.    -T X 


^^  ,^,H      d^b  '  a'^     ^2 
a—1     a—1     a+S     a+S 


29. 


30. 


a+4 


a+2     a+2  '  ^--2     a--2 
a    "^«— 3        3    "^«— 1 

a  a 


a  a 


180.  When  an  operation  is  performed  upon  a  poly- 
nomial, some  or  all  of  whose  terms  are  fractions,  we 
naturally  combine  all  the  terms  into  a  single  fraction 
and  then  perform  the  indicated  operation.  Sometimes, 
however,  the  operation  may  be  performed  without  thus 
combining. 

a^     a     \         a     1 
For  instance,  if  we  wish  to  multiply  -77-+ 0+7  t>y  o~~o'  ^^  would 

by  the  previous  process  combine  each  of  these  expressions  into  a 
single  fraction  as  follows: 

2^     a     1     6^2  +  4^  +  3 


and 


2'*'3^4~         12 
a     l_3a-2 
2    3~     6 

6^2  +  4^  +  3^  3^-2 


[a*     a     l\[a     11     6^2  +  4^  +  3, 
Therefore.        [_+-+-|  [---J  =_^^-X 


72" 


FRACTIONS.  117 

But  we  may  also  multiply  these  expressions  together  in  the  same 
way  as  integral  expressions  are  multiplied,  as  follows: 

1 


2'^3"^4 

a     1 

2     3 

a^     a 

1 

~"6"~9" 

"12 

a^      a      a 

1 

T"^  8~9~ 

"12 

The  terms  of  this  result   may  be  combined,  after  reducing  to  a 

common  denominator,  72.     Combining  terms, 

a^     a     a^_\  _18g3 +  9^-8^— 6_18^^ +^-6 
T'^8~9~12""  72  ~         72 

which  is  the  same  as  was  reached  by  the  other  process. 

As  another  illustration,  let  us  divide  -^ ttft-^-^  by  o— s- 

o        do       4         o     J 


The  work  may  be  arranged  as  follows: 

a     \\a^     13^2       1/^_^_1 
3~2;"6"      36"  "^iVT    3~2 


"T 


a^  1 

--TT        +7 


9  "^6 


z      1 

5  +  4 


The  terms  of  this  quotient  may  be  combined,  giving 
3^2_2^— 3 
6 
which  is  the  same  result  as  would  be  obtained  by  reducing  both  divi- 
dend and  divisor  to  single  fractions  and  proceeding  by  the  method 
already  given  for  dividing  one  fraction  by  another. 


Il8  UNIVERSITY   ALGEBRA. 


KXAMPI,E)S. 


By  the  method  used  in  these  two  illustrations  work 
examples  1  to  9. 

1.  Multiply  ;t: 2  H — by  ;t: 

2.  Multiply  ^+1+2  by  1+^- 

X^        X  1  X  X 

3.  Multiply 1— r— I  by -r- 

^  '^       a       b      ab         a      0 

4.  Divide  ■g—y2-  +  i2  +  l-3^  by  2-3- 

/I         1\2  11 

5.  Divide(-+^)-lby-+^-l. 

^    ^.   .^    x^      1   ^    x^     1 

6.  Divide  jg-^  by -J---. 

;i;2  4      1  1 

7.  Divide  ;»; 3 — o-+-^"~o ^byjr+  — 

27  3 

8.  Divide  8x^-j — 3  by  2x-i-—  and  multiply  the  result 

by^+i. 

9.  Multiply  ;ir^ ^  by  ;rH —   and   divide   the   result 

by  X . 

X 

x-^y y-\-2 z+x        ^       ' 

•  (^-^)(^-j^)     (^-ji/)(^— ^)     {y—2){y—x)~''^ 
Simplify  each  of  the  following : 


12. 

13 

x^^^-l 

x^- 

~1 

x^- 

X 

X— 

-1 

x^-1    * 

X- 
-X 

V 

-1 

x'' 
x^ 

■1 

1_1 

14. 


a+ 


CHAPTER  IX. 

POWKRS   AND   ROOTS. 

181.  The  process  of  raising  any  number  or  expression 
to  any  power  is  called  Involution. 

182.  In  Art.  121  we  learned  that  the  product  of  any 
number  of  factors,  each  of  which  is  a  power  of  the  same 
number,  is  equal  to  that  number  raised  to  a  power  whose 
i?idex  is  the  sum  of  the  indices  of  the  factors.  In  Art.  122 
we  learned  that  the  rth  power  of  the  nth  power  of  a  num- 
is  equal  to  the  rnth  power  of  that  number.  In  Art.  123 
we  learned  that  the  n  th  power  of  the  product  of  any  num- 
ber of  numbers  is  equal  to  the  product  of  the  n  th  powers  of 
those  numbers.  Also  that  the  n  th  power  of  the  quotient 
of  two  numbers  equals  the  quotient  of  the  nth  powers  oj 
those  numbers. 

Thus  we  may  raise  to  any  desired  power, 

First,  a  power  of  a  number; 

Second,  the  product  of  several  numbers ; 

Third,  the  quotient  of  two  numbers. 

These  three  cases  include  every  monomial  that  can  be 
proposed.  Hence  any  monomial  can  be  raised  to  any 
required  power  by  the  methods  already  given. 

183.  As  any  even  power  of  a  monomial  having  a  — 
sign  is  the  product  of  an  even  number  of  subtractive 
terms,  it  follows  that  any  even  power  of  a  monomial 
having  a  —  sign  is  a  monomial  having  a  +  sign  or  no 
sign.  Thus,  (-2)2  = +4,  (-3)4= +81,  (-2^^2)6=, 
64^6^1  ^  etc. 


I20  UNIVERSITY   ALGEBRA. 

184.  Again,  as  any  odd  power  of  a  monomial  having 
a  —  sign  is  the  product  of  an  odd  number  of  subtractive 
terms,  it  follows  that  any  odd  power  of  a  monomial 
having  a  —  sign  is  a  monomial  having  a  —  sign.    Thus, 

e;xampi,e)s. 

Write  down  the  values  of  the  following  expressions : 
/2£2£\8  /5£y\3  /      4^2^  \^ 

^*  KhmrV  ^'  \  2uv  )'  ^*  V     u'^vzaV' 

^'  \2^j)'  ^'  \   2uvy'  '  [j^y  \y)' 

^°-  V^jrJ  Kbyz)'  ^^'  \2x''y)  '  \2xyV' 

^^   \2xz)  *•  VW'  ^^-  \xyV  '  V     rj/V* 

SQUARE   OF  A  BINOMIAI,. 

185.  In  Art.  84  we  found  that 

(^a  +  dy  =  a^+2ad+5\ 
or,  stated  in  words,  the  square  of  the  sum  of  any  two  num- 
bers is  equal  to  the  square  of  the  first,  plus  twice  the  product 
of  the  two,  plus  the  square  of  the  second, 

186.  In  Art.  84  we  also  found  that 

or,  stated  in  words,  the  square  of  the  difference  between  any 
two  numbers  is  equal  to  the  square  of  the  first,  minus  twice 
the  product  of  the  two,  plus  the  square  of  the  second. 

The  two  statements  in  italics  are  so  important  and  are  used  so  often 
that  they  should  be  thoroughly  familiar  to  the  student. 


POWERS   AND    ROOTS.  12  1 

KXAMPLKS. 

According  to  the  two  statements  in  italics,  write  down 
the  square  of  each  of  the  following  binomials : 

1.  40-3.  5.  10x+:y.  9.  (Sad^y-\-2aH. 

2.  x^—3y^,  6.  2x^+^2^  10.  (Sx^y-i-CBxy, 
X      a                    ^    2xyz  ,   2rst  m  ^^o     /  1  \  ^ 

CUBK   OF  A   BINOMIAI,. 

187.  In  Art.  84  we  found  that 

Multiplying  each  member  of  this  equation  by  «  +  3,  we  get 

(a  +  ^)^  =  ^3  +  3«2^+3a^2  +  ^3^ 

The  student  should  actually  go  through  the  work  of  multiplying 
the  second  member  of  the  jQrst  equation  by  a-\-b  to  see  that  this 
result  is  correct. 

As  a  and  b  may  stand  for  any  numbers  whatever,  we 
may  say  that 

The  cube  of  the  sum  of  a7iy  two  numbers  is  equal  to  the 
cube  of  the  first,  plus  three  times  the  square  of  the  first  mul- 
tiplied by  the  seco7id,  plus  three  times  the  first  mtdtiplied  by 
the  square  of  the  secojid,  plus  the  cube  of  the  second, 

188.  In  Art.  84  we  found  that 

ia—by^a'^—2ab^-b'^. 
Multiplying  each  member  of  this  equation  by  a—b,  we  get 
{a-bY^a^-Za''b-\-Zab''-b^, 
As  a  and  b  may  stand  for  any  numbers  whatever,  we 
may  say  that 

The  cube  of  the  difference  of  two  numbers  is  equal  to  the 
cube  of  the  firsts  minus  three  times  the  square  of  the  first 


122  UNIVERSITY   ALGEBRA. 

multiplied  by  the  second,  plus  three  times  the  first  multiplied 
by  the  square  of  the  second,  minus  the  cube  of  the  second. 

The  two  statements  in  italics  are  so  important  and  so  frequently- 
used  that  they  should  be  thoroughly  familiar  to  the  student, 

KXAMPI^KS. 

By  means  of  the  two  statements  in  italics,  write  down 
the  cube  of  each  of  the  following  binomials : 

I.  Zx-\-^'  7.  uv^ — ^.        13.  — h-- 

3  uv^  ^   a      a 

4-1+4  ^0.4-1  .^^'^'     -'^' 


a      b  '         4:X^  '     1x         In 

5.  -^ II.  — :  +  l.  17.  2^2 

ab     ac  rst  a 

^   a      b  Zx     20/^3  ^2     22 

6.  -+-  12. -^ —  18.  — 

b     a  y        X  2       a 

189.  In  Art.  92  we  learned  that  the  square  of  any 
polynomial  may  be  written  down  directly  by  writing 
first  the  sum  of  the  squares  of  each  of  the  tenets  of  the  given 
polynomial,  and  to .  this  adding  twice  the  product  of  each 
term  by  each  term  that  follows  it  in  the  given  polynomial. 

In  applying  this  method  it  must  be  remembered  that 
the  letters  we  have  used  may  stand  for  negative  as 
well  as  positive  numbers,  and  in  those  terms  which  are 
made  up  of  twice  the  product  of  each  term  by  each  term 
that  follows  it,  the  rule  of  signs  must  be  observed. 

By  the  method  just  explained  the  square  ot  a—x — z  is 
^2  +;i;2  -i-2'2  —  2a;j; — 2a2+2x2, 
the  last  term  having  the  sign  +  because  the  two  terms 
multiplied  together  to  produce  it  have  like  signs. 


POWERS    AND    ROOTS.  1 23 

KXAMPI^KS. 

Write  down  the  squares  of  the  following  polynomials : 

1.  x^-Sx'^  +  Sx-l.  9.  a^  +  C^dy  +  iScy-x. 

2.  x^ +Zx'^y+2>xy'^+y^,  10.  xy+ab—c'^+2'^. 

3.  a-\-b-\-c--d—e.  11.  abc—xyz+x'^+y'^. 

4.  2« — 2b— c — d—e.  12.  mrs'^—uv+st—w. 

5.  x+y+z—a—b—c.  13.  (a^)^— ;»;2^+(2;t:)2+j/. 

6.  u+2v-\-Sw—4:X,  14.  a  +  2:r+^;r+2a;ir2. 

8.  w  +  2r— ^^+/.  16.  a^;i;2— e^z^2_|.(;^^)2^ 

ROOTS   OF  MONOMIAI^. 

190.  The  process  of  finding  a  number  or  expression 
when  a  power  of  that  number  or  expression  is  given  is 
called  Evolution.  The  number  or  expression  found  by 
evolution  is  called  a  Root  of  the  number  or  expression 
given. 

191.  As  there  are  square,  cube,  fourth,  fifth,  etc., 
powers,  so  there  are  square,  cube,  fourth,  fifth,  etc., 
roots.  The  square  root  of  a  given  expression  means  that 
expression  which  squared  will  produce  the  given  ex- 
pression. The  Tzth  root  of  a  given  expression  means 
that  expression  which  raised  to  the  ^th  power  will  pro- 
duce the  given  expression.  For  example,  2^  =  8,  there- 
fore the  cube  root  of  8  is  2;  2^  =  16,  therefore  the  fourth 
root  of  16  is  2,  etc. 

192.  A  root  is  indicated  by  the  sign  V ,  called  a 
Radical  Sign.  A  horizontal  line  usually  extends  from 
the  upper  end  of  the  radical  sign  over  the  expression  of 
which  the  root  is  to  be  extracted.     See  Art.  27. 


124  UNIVERSITY   ALGEBRA. 

To  indicate  what  root  is  to  be  extracted,  a  small  figure 
called  the  Index  of  the  root  is  placed  in  the  angle  of  the 
radical,  except  in  the  case  of  the  square  root,  in  which 
the  index  is  not  used. 

Thus,  the  square  root  of  16  isjndicated  by  1/16,  the 
cube  root  of  8  is^  indicated  by  1^8,  the  ^th  root  of  a  is 
indicated  by  i/ a, 

A  letter  may  be  used  as  the  index  of  a  root.  Thus, 
ly a  means  the  ;^th  root  of  a,  that  is,  a  number  which 
raised  to  the  ;^th  power  will  produce  a. 

193.  We  must  notice  one  important  distinction  be- 
tween raising  to  a  power  and  extracting  a  root.  If  we 
have  given  an  expression  to  be  raised  to  a  given  power, 
we  obtain  only  one  result ;  but  if  we  have  an  expression 
given  to  extract  a  given  root,  we  may  sometimes  obtain 
more  than  one  result.  

For  example,  5^  =  25,  hence_we  say  l/25=5;  but  also 
(—5)2  =  25,  hence  we  say  l/'25=— -5. 

It  appears  thus  that  there  are  two  numbers,  +5  and 
—5,  either  of  which  is  a  square  root  of  25.  The  two 
results  are  often  written  together  by  means  of  the  double 
sign  ±.     Thus,  l/25=±5. 

194.  If  any  number  be  raised  to  any  even  power,  that 
number  is  used  an  even  number  of  times  as  a  factor,  and 
therefore  the  result  must  be  positive;  but  this  same 
result  can  be  obtained  by  raising  to  the  same  power 
as  before  the  original  number  with  its  sign  changed. 
Therefore,  any  even  root  of  a  positive  number  is  either  pos- 
itive or  negative, 

195.  If  a  number  be  raised  to  an  odd  power,  that 
number  is  used  an  odd  number  of  times  as  a  factor,  and 


POWERS    AND    ROOTS.  12$ 

therefore  the  result  is  a  number  of  the  same  sign  as 
the  one  given.  Therefore,  any  odd  root  of  a  number  has 
the  sam.e  sign  as  the  number  itself. 

196.  If  any  number  be  raised  to  an  even  power,  the 
result  \^  positive.  Therefore,  there  is  no  positive  or  nega- 
tive number  which  raised  to  an  even  power  will  give  a 
negative  result.  Therefore,  we  cannot  find  an  even  root  of 
a  negative  number.  An  even  root  of  a  negative  number 
is  called  an  Impossible  or  Imaginary  Number. 

197.  To  find  any  root  of  any  expression  we  naturally 
look  to  see  how  the  corresponding  power  was  obtained, 
and  then  go  through  the  work  backward  if  possible,  thus 
returning  to  the  expression  from  which  we  started  in  the 
case  of  involution. 

We  have  found  that  {a**y=a**'';  that  is,  a"  is  an  expres- 
sion which  raised  to  the  rth  power  gives  a*"")  therefore, 

Hence,  to  extract  the  n  th  root  of  a  power  of  an  expres- 
sion^ we  divide  the  exponent  of  the  given  power  by  the  in- 
dex of  the  root;  but  in  order  to  perform  the  division,  the 
exponent  of  the  power  must  be  a  multiple  of  the  index  of  the 
root. 

We  cannot  extract  the  square  root  of  a^  because  5,  the 
exponent  of  the  power,  is  not  a  multiple  of  2,  the  index 
of  the  root. 

198.  Root  of  a  Product.  To  find  the  nih  root  of  the 

product  of  two  factors,  we  have 

a-b''^{aby\ 

therefore,  1^'^^=  i/(aby-=ab. 

In  this  result  the  first  factor  a  may  be  found  by  taking 
the  71  th  root  of  a"",  the  first  factor  of  the  given  expression; 


126  UNIVERSITY   ALGEBRA. 

and  the  second  factor  b  of  the  result  may  be  found  by 
taking  the  nth.  root  of  b"*,  the  second  factor  of  the  given 
expression.  Therefore,  the  n\h  root  of  the  product  of  two 
factors  is  equal  to  the  product  of  the  nth.  roots  of  those 
factors. 

199.  Of  course  the  same  argument  may  be  used  with 
more  than  two  factors,  and  hence,  evidently,  the  nth  root 
of  the  product  of  several  factors  is  equal  to  the  product  of  the 
n  th  roots  of  those  factors. 

200.  Root  of  a  Quotient.  To  find  the  wth  root  of  the 
quotient  of  two  expressions,  we  have 

a""     (a^ 
¥\b) 

In  this  result  the  numerator,  a,  is  found  by  taking  the 
n\h.  root  of  the  given  numerator,  and  the  denominator,  ^, 
is  found  by  taking  the  n  th  root  of  the  given  denominator. 
Therefore,  the  nth  root  of  the  quotient  of  two  expressions 
is  equal  to  the  quotient  of  the  n  th  roots  of  those  expressions, 

EXAMPLES. 

Find  the  square  root  of  each  of  the  following  twelve 
expressions : 

„    .  9^2  49^V  ^'     ^* 

2.  ^a^x^,         5.  -2~2-  8.  -v-g.  II.  --  --. 

a^y^  y^z^  b^  d^ 

**  49a2;t2         ^   AiX^y^  b^ 


POWERS    AND    ROOTS.  127 

Find   the   cube  root  of  each  of  the  following  nine 
expressions : 
13.  SaH^c^.         16.  — 64a9;t:i^         19.  -^Qia^ 5^ -r-Sa^ , 


27  '*    27:%:^    /^  •  ^3^3^3  '^3^6 

?^6;t:l5  27     x^  ^^^^ r^^g 

^5-  -"8^*  ^^-  -8;^  64'  ^^'      "^^  *      ~^' 

22.  Find  the  fourth  root  of  a^d'^^c'^. 

23.  Find  the  square  root  of  l^a^x^y'^^,  and  then  the 
square  root  of  this  result. 

24.  Find  the  fourth  root  of  IQa^x'^y'^'^, 

25.  Find  the  cube  root  of  ^^^^^^^24^  ^^^  Xh.Qn  the 
fourth  root  of  this  result. 

26.  Find  the  sixth  root  a^^b'^'^c'^^y  and  then  the  square 
root  of  this  result. 

27.  Find  the  twelfth  root  oi  a^^b^'^c'^'^. 

28.  Find  the  square  root  of  a^^b^^c^^j  then  the  cube 
root  of  the  result,  and  then  the  square  root  of  this  second 
result. 

29.  Find  the  fifth  root  of—S2a^x^^y^\ 

30.  Find  the  seventh  root  of  12Sa'^ b'^ c^ ^ . 

l^a^x'^^y^ 

31.  Find  the  square  root  of ^  ^      * 

201.  Before  leaving  the  subject  of  roots  of  monomials, 
it  is  well  to  notice  that  what  we  have  learned  may  be 
used  to  find  the  roots  of  arithmetical  numbers  when  the 
numbers  given  have  exact  roots. 

We  resolve  the  number  into  its  prime  factors,  and  ex- 
press it  as  the  product  of  various  powers  of  these  prime 
factors,  then  divide  each  exponent  by  the  index  of  the 


128  UNIVERSITY    ALGEBRA. 

required  root.    When  the  resulting  factors  are  multiplied 
together  the  required  root  is  found. 

Suppose  we  wish  to  find  the  square  root  ol  53361. 


3 
3 

7 

7 
11 

53861=. 
1/63861 =S 

EXi 

square  root 

63861 

17787 

5929 

847 

121 

Hence, 
therefore, 

I.  Find  the 

11 
32x72xll^• 
.x7xll=231 

i.MPLES. 

of  5184. 

2.  Find  the  square  root  of  43264. 

3.  Find  the  cube  root  of  85184. 

4.  Find  the  cube  root  ot  32768. 

SQUARE   ROOT   OF   POIvYNOMIAI^S. 

202.  To  find  out  how  to  extract  the  square  root  of  a 
polynomial  we  must  see  how.  the  polynomial  was  pro- 
duced by  squaring.     We  know  that 

(x+y)  '^=x'^  +  2xy+y'^  ; 
therefore  we  know  that  the  square  root  of  x'^  +  2xy+y'^ 
is  x+y.     Our  problem,  then,  is  this :    Given  the  expres- 
sion x'^+2xy-\-y'^,  to  find  from  it  the  expression  x+y. 

The  first  term  x  of  the  root  is  the  square  root  of  the 
first  term  x'^  o^  the  given  expression.  I^et  us  set  down 
the  term  x  already  found,  and  subtract  its  square  from 
the  given  expression.  There  remains  of  the  given  ex- 
pression 2;t:j/-j-j/2  or  (2x-\-y)y. 

From  this  we  see  that  the  second  term  y  of  the  root 
will  be  the  quotient  when  the  remainder  just  found  is 


POWERS   AND    ROOTS.  1 29 

divided  by  2x-\-y.     This  divisor  2x+j/  consists  of  two 
terms,  the  first  of  which  is  twice  the  portion  of  the  root 
already  found,  and  the  second  is  the  new  term  y  itself. 
The  work  may  be  arranged  as  follows: 

x'^-i-2xy+j/^  (  x+j/ 
x^ 


2x+y 


2xy-\-y^ 
2xy-\-y'^ 


After  the  first  term  x  of  the  root  has  been  found,  its 
double  2x  is  used  as  a  trial  divisor  by  which  to  divide 
the  remainder  '^xy+y'^.  We  see  that  the  first  term  2xy 
of  this  remainder,  when  divided  by  the  trial  divisor  2:r, 
gives  y,  from  which  we  judge  that  y  is  the  next  term  in 
the  root.  When  the  y  is  thus  found,  it  is  added  to  the 
trial  divisor  2;r,  giving  the  complete  divisor  2x+y,  and 
this  is  multiplied  by  r,  giving  the  expression  2xy+y'^. 

203.  Of  course  we  may  obtain  by  this  process  the  dif- 
ference of  two  numbers  for  our  square  root  as  well  as  the 
sum,  as  in  the  example  just  given.    This  will  be  plain  by 
u        working  out  another  example. 

To  find  the  square  root  of  Aa^  —  12ad+ d^ , 
Arrange  the  work  thus : 

4^2 

iaSd  i  -12a5-\-9d^ 
I  -12ad  +  9d'^ 

Here  the  first  term  of  the  remainder  — -12«^,  when 
divided  by  the  trial  divisor  Aa,  gives  the  quotient  —Sd. 
Hence  we  judge  that  --3^  is  the  next  term  of  the  root, 
and  upon  trial  this  proves  to  be  right. 


L 


9  — U  A. 


I30  UNIVERSITY    ALGEBRA. 

KXAMPI^KS. 
Find  the  square  root  of  each  of  the  following : 

1.  a^-{-4ab  +  4:d\     5.  x^  +  2x^+x'^.       9.  a'^  —  2aH^  +  d^. 

2.  4a'^—4ad+d\     6.  x^  —  2x^+x^,     10.  a^  —  2ad^+d\ 
S.  4:a^—Sad+Ad\   7.  x^—2x^+x^,     11.  a^  +  2a^d'^ +  d^, 
4.  4^2— 16«+16.     8.  x'^  +  2x'^-\-l.       12.  9x^  —  18x'^  +  d, 

13.  a^—2a^x^+x^,  16.  Ax^  —4:nx^j/-i-n^y'^ , 

14.  a'^d^  +  2adcd+c'^d^,      17.  a2^4_2^^;^;3  +  ^2^2^ 

15.  a4^*-6a2^2_^9.  18.  a^d^-2a''d^c^+c^\ 

204,  Thus  far,  the  polynomials  of  which  we  have  ex- 
tracted the  square  root  have  been  in  every  case  those  of 
three  terms.  The  above  process,  however,  can  be  ex- 
tended so  as  to  find  the  square  root  of  any  polynomial 
which  is  a  perfect  square,  no  matter  how  many  terms  the 
polynomial  contains.  For  example,  to  find  the  square 
root  ofa'^+d^+c'^-i-2ad+2ac-^2dc. 

First  arrange  the  expression  according  to  powers  of 
some  letter,  say  a,  and  write  a"^  -\-2a3  +  2ac+  d^  +  2dc-i-c^ . 
The  first  term  ^^  of  this  polynomial  is  produced  by 
squaring  a.  Therefore  the  first  term  of  the  root  is  a, 
and  the  whole  root  is  <3^-f  something,  and  this  something 
is  what  we  wish  to  find. 

Proceeding  as  before  with  the  first  term  of  the  root,  a 
part  of  the  process  may  be  arranged  thus : 

a^+2ad+2ac-i-d'^  +  2dc+c^  (  a 

a^ 

2ab+2ac+b'^+2bc+c'^ 

Now  twice  a  used  as  a  trial  divisor  would  suggest  b  for 
the  next  term  of  the  root.  Call  the  next  term  b  and  pro- 
ceed as  before,  and  the  work  will  stand  thus : 


POWERS    AND    ROOTS.  I31 

^2 


2a  +  d 


2ab-^2ac-^d'^+2dc+c^ 
2ab  +<^2 


2ac         +2bc+c'^ 

There  is  sfzll  a  remainder,  so  we  have  not  yet  found 
the  entire  root;  but  the  root  is  <t  +  ^+ something,  and  this 
something  is  what  we  wish  to  find. 

Now  let  us  consider  a-\-b,  the  part  of  the  root  already 
found,  as  a  single  term,  and  use  it  as  we  have  before  used 
the  first  term  of  the  root.  We  must  then  take  twice  a-\-b 
and  use  it  as  a  trial  divisor  by  which  to  divide  the  last 
remainder  2ac-{-2bc+c'^,  from  which  we  judge  that  c  is 
the  next  term  of  the  root.  When  the  c  is  thus  found,  it 
is  added  to  the  irml  divisor  2a  +  2b,  giving  the  complete 
divisor  2a-i-2b-\-c,  and  ^Ms  is  multiplied  by  c,  giving  the 
expression  2ac-{-2bc-j-c^. 

The  work  from  the  beginning  will  now  stand  thus : 
a'^+2ab+2ac+b^  +  2bc-i-c^  (  a  +  b+c 

/,2 


2a  +  b 


2ab+2ac-j-b^+2bc-{-c^ 
2ab  -f-<^2 


2a-\-2b+c       2ac         ■i-2bc-\-c^ 
2ac         -{-2bc+c^ 

As  there  is  now  no  remainder,  the  process  is  ended  and 
the  root  is  a  +  b-\-c.  If  after  finding  the  third  term  of  the 
root  there  were  still  a  remainder,  we  would  group  the 
three  terms  thus  found  into  a  single  term,  and  use  this 
group  as  we  have  always  used  the  first  term  of  the  root. 

The  process  may  evidently  be  extended  to  finding  the 
square  root  of  any  polynomial  that  is  a  perfect  square. 

205.  From  what  has  been  said  we  can  see  that  the 
Method  of  Finding  the  Square  Root  of  any  Poly- 
nomial is  as  follows: 


132  UNIVERSITY    ALGEBRA. 

I.  Arrange  the  terms  according  to  powers  of  some  letter. 

II.  Find  the  square  root  of  the  first  term,  write  it  as  the 
first  term  of  the  root,  and  subtract  its  square  from  the  given 
expression. 

III.  Use  twice  the  portion  of  the  root  already  found  as  a 
trial  divisor,  and  divide  the  remainder  just  found  by  this 
trial  divisor.  Add  the  quotient  to  the  first  term  of  the  root 
and  also  to  the  trial  divisor. 

IV.  Multiply  the  complete  divisor  by  the  term  of  the  root 
last  obtained,  and  subtract  the  product  from  the  remainder^ 

V.  Repeat  III  and  IV  until  there  is  no  remainder, 

'    EXAMPLES. 

Find  the  square  root  of  each  of  the  following : 

1.  x^+4:xy+2xz-\-4y^+4y2-\-^^. 

2.  x^—6xy—4:XZ-\-9y'^  +  12yz+4:^'^, 

3.  4:x'^-^16xy—Ax2+16y'^—Sy^+3^. 

4.  a''-2aH-{-2a''c+b'--'2bc-{-c\ 

5.  4a^-12aH^-8a^c+9b^  +  12b^c+4:c\ 

6.  a^  +  2aH^+2a^c''-\-b^  +  2b^c^+c\ 

7.  a'^ -i-2ab  +  2ac+2ad+b'^  +2bc-\-2bd+c^  +2cd+dK 

8.  a^-^-iab—eac+W^—nbc+dc^ 

9.  a^+2a^  +  Sa^  +  4:a^  +  Sa^--i-2a  +  l. 
10.  9x^  +  lSa'^x'^-\-6x^  +  9a^  +  6a'^-Jrl. 

CUBE   ROOT   OF  POI^YNOMIAI^. 

206.  To  find  how  to  extract  the  cube  root  of  a  poly- 
nomial we  must  see  what  the  cube  of  an  expression  is, 
and  then  return,  if  possible,  from  the  cube  to  the  expres- 
sion from  which  this  cube  was  obtained. 

207.  We  know  that  (x+yy=x^-hSx^y-\-Sxy'^+y^. 
Our  problem,  therefore,  is  this:     Given  the  expression 


POWERS   AND    ROOTS.  1 33 

x^-\-Sx'^j/+Sxy'^+j/^,    to    find    from   it   the   expression 
x-^y,  which  is  the  cube  root  of  the  given  expression. 

208.  We  see  that  the  first  term  x  of  the  root  is  the 
cube  root  of  x^,  the  first  term  of  the  given  expression. 
Set  down  the  x  as  the  first  term  of  the  root  and  subtract 
its  cube  from  the  given  expression,  and  we  have  a  remain- 
der Sx'^y-j-Sxj/^-^y^  or  (Sx^  +Sxy-i-y'^')y.  We  see  from 
this  that  the  second  term  y  of  the  root  is  the  quotient 
obtained  by  dividing  this  remainder  by  Sx'^+Sxy-^y'^. 

Now  this  divisor  is  composed  of  three  terms,  of  w^hich 
the  first  is  three  times  the  square  of  the  first  term  of  the 
root,  the  second  is  three  times  the  product  of  the  first 
term  of  the  root  and  the  new  term  y,  and  the  third  is  the 
square  of  the  new  term  of  the  root.  The  sum  of  thCvSe 
three  terms  constitute  the  complete  divisor,  while  the 
remainder  found  by  subtracting  x^  from  the  given  ex- 
pression is  the  complete  dividend. 

This  complete  divisor  contains  two  terms  which  involve 
the  jK,  which  is  not  yet  supposed  to  be  known.  However, 
we  may  get  something  of  an  idea  of  what  the  second  term 
of 'the  root  must  be  by  using  the  Jirs^  term  of  the  above 
remainder  as  a  trial  dividend,  and  three  times  the  square 
of  the  first  term  of  the  root  as  a  trial  divisor,  and  then  it 
the  number  we  get  by  this  division  is  correct,  the  com- 
plete divisor  (which  can  then  be  found)  when  multiplied 
by  the  new  term  of  the  root  must  give  the  complete  div- 
idend, i.  ^.,  the  remainder. 

209.  The  work  may  be  arranged  as  follows : 

x^  -{-Sx^y+Sxy'^  -{-yl  (  x+y 
x^ 


Sx'^+Sxy+y'' 


Sx'^y+Sxy'^+y''^ 
Sx'^y+Sxy'^+y^ 


134  UNIVERSITY   ALGEBRA. 

210.  The  remark  made  under  square  root  (Art.  203) 
about  the  —  sign  applies  here  as  well,  as  is  illustrated  in 
the  following  example : 

Find  the  cube  root  oi  x^  —  Qx'^y+12xy^  —  Sy^, 

The  work  is  as  follows : 


X 

Sx'^-6xy+4:y^ 


x^  —  6x^y+12xy—8y^  (  x~2y 

3 


—6x^y+12xj/^—8y^ 


i^XAMPLES. 


Find  the  cube  root  of  the  following  expressions : 

1.  x^y^-}-12x^y^-h4Sxy^-^6Ay\ 

2.  8a^—60a'^x+150ax'^—125x^. 

3.  27x^—5Ax^y2-i-S(jx'^y'^2'^—Sy^2^. 

5.  SaH^-'2Sa'^d^c+24adc'^-8c\ 

211.  So  far  all  the  expressions  of  which  the»cube  root 
was  required  were  polynomials  of  four  terms,  but  we  may 
have  a  polynomial  of  more  than  four  terms  of  which  the 
cube  root  is  required.  In  this  case  the  process  already 
given  may  be  extended,  as  in  the  case  of  the  square  root, 
viz. :  Find  two  terms  of  the  root,  as  already  explained, 
and  then  consider  these  two  terms  as  a  single  term  and 
use  their  sum  the  same  as  a  single  term  was  used  before. 

EXAMPLKS. 

Find  the  cube  root  of  each  of  the  following : 

1.  a^  +  Sa^+6a^-{-7a^+6a^  +  Sa  +  l. 

2.  x^—6x^A-9x^-\-4x'^-9x^-6x-l. 

3.  64:X^  +  192x^-hU4x^-S2x^S6x''-{-12x'-l. 


CHAPTER  X. 

SIMPI.K   EQUATIONS. 

212.  An  Equation  is  the  statement  of  equality  which 
exists  between  two  expressions.     See  Arts.  23  and  24. 

213.  The  Members  or  Sides  of  an  equation  are  the 
parts  on  either  side  of  the  sign  = ,  and  are  distinguished 
as  the  First  and  Second  Members,  or  Left  and  Right 
Sides,  respectively. 

Students  often  have  a  careless  habit  of  calling  almost  everything 
in  Algebra  an  equation.  Thus  we  hear  a^  +  ^ab-^-b"^  called  an  equation 
instead  of  an  expression  or  a  trinomial.  It  is  better  to  call  an  expres- 
sion an  expression,  and  an  equation  an  equation. 

214.  If  the  two  sides  of  an  equation  are  equal,  no 
matter  what  numbers  are  substituted  for  the  letters,  the 
equation  is  called  an  Identical  Equation  or  simply  an 
Identity.     Thus  the  following  equations  are  identities: 

x[x — a)=ji:^ — ax\ 
(x-\-a)(x—a)=x^  —a^  ; 
for  the  equations  are  true,  no  matter  what  numbers  are 
put  for  x  and  a, 

215.  If  the  two  sides  of  an  equation  are  equal  only 
when  particular  numbers  are  substituted  for  the  letters 
the  equation  is  called  a  Conditional  Equation  or  simply 
an  equation.    The  following  are  conditional  equations: 

.r+l  =  2; 

for  the  first  equation  is  true  only  when  .r=l,  and  the 
second  is  true  only  when  x^=Z, 


136  UNIVERSITY   ALGEBRA. 

216.  A  letter  for  which  a  particular  number  must  be 
substituted  in  order  that  the  two  sides  of  the  equation  may 
be  equal  is  called  an  Unknown  Number.     See  Art.  42. 

217.  A  number  which,  when  substituted  for  the  un- 
known number,  makes  the  two  sides  of  the  equation 
equal  is  said  to  Satisfy  the  equation,  and  is  called  a 
Root  of  the  Equation  or  a  Value  of  the  Unknown 
Number. 

218.  To  Solve  an  equation  is  to  find  the  root  or  roots. 

219.  A  Simple  Equation  or  Equation  of  the  First . 
Degree  is  one  which  when  reduced  contains  only  th^Jirs^ 

X-4-S 
power oi  the  unknown  nu^nber.     Thus,  '^—\x — 4;r=— j— 

is  a  simple  equation,  but  x'^  -^^x=^h  is  not  a  simple  equa- 
tion. Simple  equations  are  also  called  Linear  Equations. 

220.  Axioms.  The  following  self-evident  truths,  or 
Axioms,  are  made  use  of  in  solving  equations: 

I.  If  we  add  to  equals  the  same  number  or  equal  numbers 
the  sums  will  be  equal, 

II.  If  we  take  from  equals  the  same  number  or  equal 
numbers  the  remainders  will  be  equat. 

III.  If  we  7miltiply  equals  by  the  same  numbers  or  equal 
numbers  the  products  will  be  equal. 

IV.  If  we  divide  equals  by  the  same  number  or  equal 
numbers  the  quotients  will  be  equal, 

221.  The  use  of  the  axioms  is  illustrated  by  the  fol- 
lowing examples : 

(1)  Solves -19=32. 

We  have  given  5C— 19=32 

Adding  equals  to  both  members  (Axiom  1),  19     19 

Whence,  x—^\ 


SIMPLE    EQUATIONS.  137 

(2)  Solve  5^4-12=87. 

We  have  given  5x+ 12=87 

Subtracting  equals  from  both  members  (Axiom  2),         12     12 

5x=15 
Dividing  both  members  by  5  (Axiom  4),  :r=15 

3x 

(3)  Solve  the  equation  -=-=9. 

Multiplying  both  sides  of  the  equation  (Axiom  3)  by  7,  we  have 

3:JCrr63. 

Dividing  both  sides  of  the  equation  (Axiom  3)  by  3,  we  obtain 

x=21. 

222.  Consider  the  equation 

x-\-a—d=c,  (1) 

Subtracting  a  from  the  members,  we  get 

X — d=c—a.  (2) 

Adding  d  to  the  members,  we  get 

x=^c—a  +  d.  (3) 

Comparing  (2)  and  (3)  with  (1),  we  observe 

By  the  addition  or  subtraction  of  equals  we  may  cause  a 
term  to  disappear  from,  one  mernbe?  of  aji  equation  and 
to  appear  with  its  sign  changed  in  the  other  member, 

223.  Transposition.  If  we  remove  a  term  from  one 
member  of  an  equation  and  make  it  appear  in  the  other 
member,  we  are  said  to  Transpose  that  term.  If  we  use 
this  word  we  may  restate  the  above  principle  as  follows : 

Any  term  in  one  membe7  of  an  equation  may  be  trans- 
posed to  the  other  member  provided  its  sign  be  changed. 

In  this  way  of  speaking  we  are  apt  to  keep  in  mind  merely  the 
change  which  results  in  the  equation  and  to  lose  sight  of  the  addition 
or  subtraction  of  equals  which  causes  transposition.  Axioms  1  and  2 
must  always  be  appealed  to  when  we  are  called  upon  to  explain  why 
transposition  is  allowable. 

(1)  Solve  1  +  3+ 3=^ -'7. 

Multiplying  both  sides  of  the  equation  (Axiom  3)  by  6,  we  have 

6  +  3x+2xz=6x;-42. 


138  UNIVERSITY   ALGEBRA. 

Transposing  (Axioms  1  and  2)  unknown  numbers  to  the  left  side  and 
known  numbers  to  the  right  side,  we  get 

Sx+2x-Qx=— 4:2-6, 
Uniting  similar  terms,  —  ^=:— 48. 

Dividing  both  sides  (Axiom  4)  by  - 1,  we  have 

^=48. 

(2)Solve^-^+^=2--+5f. 

Multiplying  both  sides  of  the  equation  (Axiom  3)  by  12,  we  have 

Qx-4x-j-3x=24:-2x  +  5x, 
Transposing  unknown  numbers  to  left  side, 

Qx  —  4iX  +  3x-\-2x-ox= 24. 
Uniting  similar  terms,  2^=24. 

Dividing  both  sides  (Axiom  4)  by  2,  we  have 

5C=:12. 

224.  Verification.  As  mistakes  are  sometimes  made 
in  solving  an  equation,  it  is  well  for  the  student  to  test 
the  result  by  substituting  the  value  found  for  the  un- 
known number  in  place  of  the  letter  representing  it  in 
the  original  equation.  If  the  equation  thus  found  is  not 
true  a  mistake  has  been  made  and  the  solution  should  be 
re-examined.  This  process  of  testing  a  result  is  called 
the  Verification  ot  that  result. 

EXAMPIvES. 

Solve  each  of  the  following  equations : 
I.  5x-2=Sx+18, 

SOLUTION. 

Transposing  the  terms  3x  and  -  2, 

5x-3;c=184-3. 
Uniting  terms,  2x:=20. 

Whence,  5C=10. 

VERIFICATION. 

Substituting  10  for  x  in  the  original  equation, 
5X10-2=3x10+18, 
or  50-2=30  +  18; 

and  since  this  equation  is  true  the  correct  value  of  x  has  been  found. 


SIMPLE    EQUATIONS.  139 

2.  6(x-5')+2x=8x—2(x-{-10). 

Performing  the  multiplication  by  6  and  —2,  we  have 
6x-30+2x=8x-2;c-20. 
Transposing  the  terms  —30,  8x,  and  —2x,  we  obtain 

Qx-\-2x-Sx+2x-S0-20, 
Uniting  similar  terms,  2^=10. 

Whence,  x^5. 

3.  9(13-^)-4;i:=5(21-2^)+9;r. 

4.  7(3^-6)  +  5(:r--3)+4(17-;»;)=ll. 

5.  2(16-x)-hS(i5x-4:)  =  12(S+x')-'2(12-x) 

225.  The  object  in  multiplying  both  sides  of  an  equa- 
tion by  the  same  number  is  to  Clear  the  Equation  of 
Fractions.  This  may  be  accomplished  in  two  ways: 
First,  dy  multiplying  both  members  of  the  equation  by  the 
product  of  the  denominators  of  all  the  fractions;  or,  if  we 
prefer,  by  multiplying  both  members  of  the  equation  by  the 
least  common  denominator  of  alt  the  fractions, 

%x    llx     Wx 
Thus,  solve  the  equation  -7-  +  — H — -pr  -366=0. 
4      lo       6 

Multiplying  both  sides  by  60,  the  least  common  denominator  of 
the  fractions,  A5x+2Sx-^  llO;*:- 21960=0. 

Transposing  known  number  to  right  side  and  uniting  similar  terms, 

183x=21960. 
Dividing  both  members  by  183,  we  obtain 

.^=120. 

226.  In  clearing  an  equation  of  fractions  |the  student 
must  take  special  care  when  a  minus  sign  occurs  before 
a  fraction  and  the  numerator  is  not  a  monomial.  It 
must  be  remembered  that  the  dividing  line  of  a  fraction  is 
the  same  as  a  parenthesis  enclosing  the  numerator.     Thus, 

x-\-4c.  x-\-^ 

2 T—  is  the  same  as  2~(.r+4),  and  2 =—  is  the 

1  o 

same  as  2—\{x-\-^.     We  will  illustrate  this  point  by  a 
few  examples. 


140  UNIVERSITY   ALGEBRA. 

(1)  Solve -^=-^^ 2""^ 

Multiplying  both  sides  by  12, 

4{x-j-2)=3{D-x)-6{x  +  l)  +  lQS. 
Performing  the  indicated  operations, 

(4x4-8)-=  (15 -3^) -(6^ +  6) +168. 
Removing  parentheses, 

4^^+8=15 -3x-6x- 6  + 168. 
Transposing  the  known  numbers  to  the  right  side  and  the  unknown 
numbers  to  the  left  side, 

4^+3?c4-6x=15-6+138-8. 
Uniting  similar  terms,  13?c=169. 

Whence,  x=lS. 

(2)  Solves 5~"~^ 3~- 

Multiplying  both  sides  by  15, 

15^-3(3x-2)=45-5(2;r-5). 
Performing  indicated  operations, 

15;t:-(9;«:-6)=45- (10^-25). 
Removing  parentheses,      15^  — 9:x;  +  6=45  — lOjir  +  25. 
Transposing  the,  known  numbers  to  the  right  side  and  the  unknown 
numbers  to  left  side,  15.^-9jir+10:x;=45  +  25-6. 

Uniting  similar  terms,  16^=64;  .  *.  x=4:. 

227.  The  signs  of  all  the  terms  in  the  mcmerator  of  each 
fraction  preceded  by  the  minus  sign  must  be  changed  when 
the  equation  is  cleared  of  fractions.  The  neglect  of  this  is 
a  very  common  source  of  error, 

228.  Directions  for  Solving  Simple  Equations. 
The  following  is  usually  the  best  course  to  pursue  in 
solving  simple  equations: 

I.  Clear  the  equation  of  fractions  and  perform  the  indi- 
cated operations. 

II.  Transpose  the  terms  containing  the  known  numbers 
to  the  right  side  and  the  terms  containing  the  unknown 
numbers  to  the  left  side  of  the  equation. 


SIMPLE    EQUATIONS.  14I 

III.  Unite  similar  terms. 

IV.  Divide  both  sides  by  the  coefficient  of  the  unhiown 
nnmber. 

This  is  generally  the  best  order  to  pursue,  although  it  is  some- 
times shorter  to  unite  similar  terms  before  clearing  of  fractions. 

KXAMPLKS. 

Solve  the  following  equations : 
y    5j/+4_4)/-9  2(7r-10)     ^r,J>^-y 

3r+9_5y+16  _^^5     284--^         _ 

2-4-7  7.     4-         5     +br-U. 


3. 


h—y_\\—Zy  5y— 6     By— 6     4j/ 

"7    ~     23     *  -7  9         ir 


2j/-6     3>/_,y-4  3.^      ^.\-ly 


11.  ^(^-2)-K^~3)-f  K^-4)=4. 

12.  3(^+3)2+6(^+5)2=8(^+8)2. 

13.  K^-3)-K^-8)  +  i(^-5)=0. 

^'   3        4j/  12 

I^ITKRAI,  EQUATIONS. 

229.  If  the  known  numbers  in  an  equation  are  repre- 
sented by  letters  instead  of  by  figures,  the  equation  is 
spoken  of  as  a  Lriteral  Equation.  Of  course  the  same 
principles  hold  in  the  solution  of  such  equations  as  in 
the  solution  of  numerical  equations. 

It  must  be  remembered  that  the  first  and  intermediate  letters  of 
the  alphabet  stand  for  numbers  supposed  to  be  known  or  given. 


142  UNIVERSITY   ALGEBRA. 

EXAMPI^KS. 

Solve  the  following  literal  equations: 

ax—b     bx-Yc 

I. =^abc. 

c  a 

Multiplying  both  sides  by  ac,     a{ax  -b)-c  {bx-\- c)=sa*dc*. 

Removing  the  parentheses,  a^x~ad—l>cx—c^=:a*dc^. 

Transposing  known  numbers  to  right  side, 

Uniting  with  a  parenthesis,  {a^—dc)x=a^dc^  +  ad-\-c^. 

Dividmg  both  sides  by  {a^  —  oc),     x= — r— ^ — 

a^—bc 

2.  {b—V)y=b—y.  4.  ad—dy=-my—am. 

3.  {y.  —  b^x^a—x.  5.  p{x—V)+x=^q—p. 

6.  {b+V)X'\-ab=b{a^-x^-\-a. 

7.  {a-\-b^x+{a—b^x—a'^, 

8.  {a  +  bx'){b^ax)=^ab{x'^—V). 

9.  {x-Ya\x+b)^{x'--a){X'--b)-Y{a+by. 

10.  ax{x-\-a^  +  bx(jK-{-b^  =  (jx-\-b)(^x-\-a){x-\-b). 

11.  -rx^, — x=^b'^+x'^, 
b        a 

Zb(^x-a)     {x—b''')^     (^4:a+cx)b 

5a  15b  6a 

nx     r — X  ,  n(r—x) 

^3-  -y-^f"^-^--' 


14. 
15. 


2ax--2b     ax— a ax     2 

b+c       _a     ^  ,  a(a-l)^b{b-r)+c 

XX  X 

SYMBOI.IC   EXPRESSIONS. 


230.  In  solving  a  problem  in  Algebra  we  must  not 
only  select  some  letter  to  stand  for  the  unknown  number, 
but  we  are  required  to  find  expressions  containing  the 


SIMPLE    EQUATIONS.  143 

unknown  letter  which  wall  symbolize  all  other  numbers 
occurring  in  the  problem.  Such  may  be  called  Symbolic 
Expressions. 

Thus,  if  the  sum  of  two  numbers  is  100,  and  if  x  stands 
for  one  of  them,  then  the  expression  100— x  stands  for 
the  other  number.  If  x  is  the  price  of  one  horse,  then 
lOx  stands  for  the  cost  of  10  horses ;  if  5  yards  of  cloth 
cost  X  dollars,  then  the  cost  of  one  yard  is  represented  by 

X 

the  expression  ^,  etc.  Drill  in  the  formation  of  symbolic 
expressions  is  of  help  in  the  solution  of  problems. 

1.  What  two  numbers  differ  from  100  by  7?  What  two 
numbers  differ  from  100  by  ^  ?  What  two  numbers  differ 
from  nhy  x"^, 

2.  A  train  traveled  at  the  rate  of  20  miles  per  hour  for 
3  hours;  how  far  did  it  go  ?  A  train  traveled  at  the  rate 
of  r  miles  per  hour  for  3  hours;  how  far  did  it  go  ?  A 
train  traveled  at  the  rate  of  r  miles  per  hour  for  t  hours; 
how  far  did  it  go  ? 

3.  The  rate  of  a  train  is  r.  How  far  will  it  go  in 
time  /? 

4.  A  man  can  do  a  piece  of  work  in  10  days.  How 
much  of  it  can  he  do  in  one  day  ?  A  man  can  do  a  piece  of 
work  in  n  days.     How  much  of  it  can  he  do  in  one  day  ? 

5.  One  pipe  will  fill  a  cistern  in  a  hours,  and  another 
pipe  will  fill  it  in  b  hours.  What  part  does  each  pipe 
carry  in  one  hour?  What  part  do  both  pipes  together 
carry  in  one  hour  ? 

6.  The  digit  5  stands  in  tens'  place;  what  number  is 
expressed  ?  The  digit,  represented  by  x,  stands  in  tens* 
place;  what  number  is  expressed  ?  A  digit  represented 
by  x  stands  in  hundreds'  place;  what  number  is  expressed? 


144  ,  UNIVERSITY   ALGEBRA. 

7.  A  can  do  a  piece  of  work  in  9  days  and  B  can  do  it 
in  12  days.  What  part  can  each  do  in  one  day  ?  What 
part  can  both,  working  together,  accomplish  in  one  day  ? 
A  can  do  a  piece  of  work  in  a  days  and  B  can  do  it  in  b 
days.  What  part  can  both,  working  together,  accom- 
plish in  one  day  ? 

8.  The  three  digits  of  a  number  beginning  at  the 
right  are  represented  by  x,  x+2,  and  xS;  what  is  the 
number  expressed  by  them  ? 

9.  Write  three  consecutive  even  numbers  of  which  6 
is  the  first.  Write  three  consecutive  even  numbers  of 
which  2n  is  the  first. 

10.  Write  three  consecutive  odd  numbers  of  which  5  is 
the  first.  Write  three  consecutive  odd  numbers  of  which 
2n  +  l  is  the  first. 

11.  Write  three  consecutive  numbers  of  which  n  is  the 
greatest.  Write  three  consecutive  even  numbers  of  which 
2n  is  the  greatest.  Write  three  consecutive  odd  numbers 
of  which  2n+l  is  the  greatest. 

12.  Show  that  the  squares  of  any  two  consecutive 
numbers  differ  by  an  odd  number. 

13.  Show  that  if  n  stands  for  a  whole  number,  then 
nCn  +  Y)  is  divisible  by  2. 

14.  Show  that  if  n  stands  for  a  whole  number,  then 
n(Sn—l)  is  divisible  by  2. 

231.  It  is  well  to  note  that  if  n  stands  for  any  whole 
number,  then  27^  is  the  symbolic  expressions  for  any  even 
number,  since  it  is  exactly  divisible  by  2,  and  2n-\-l  is 
the  symbolic  expression  for  any  odd  number,  since  when 
divided  by  2  the  remainder  is  1. 


SIMPLE    EQUATIONS.  145 

PROBI^KMS. 

232.  Directions  for  Solving  Problems.  The  stu- 
dent will  find  that  the  following  is  the  usual  course  pur- 
sued in  solving  problems  by  Algebra : 

I.  Represent  one  of  the  unknown  numbers,  preferably 
the  one  whose  value  is  asked  for,  by  x. 

II.  Make  symbolic  expressio?is  to  represent  each  of  the 
other  unknown  numbers  mentioned  in  the  problem. 

III.  Find,  from  the  problem,  two  of  these  symbolic  ex- 
pressions  that  are  equal  to  each  other. 

IV.  Solve  the  equation  thus  formed. 

Solve  each  of  the  following  problems : 

1.  The  sum  of  two  numbers  is  33  and  their  difference 
is  7.     Find  the  numbers. 

Let  5C— one  of  the  numbers; 

then  33  — ^=the  other  number. 

And  since  the  difference  of  the  two  numbers  is  7;  therefore, 

5C-(33-;r)=7. 
Removing  parenthesis,  ^—33  +  ^=7. 

Transposing  and  uniting  terms,        2x=40; 
whence,  ^=20. 

Therefore,  one  number  is  20  and  the  other  13. 

2.  In  a  yard  there  are  chickens  and  rabbits,  and  alto- 
gether they  have  14  heads  and  38  feet.  How  many  rabbits 
and  how  many  chickens  in  the  yard  ? 

3.  A  man  was  engaged  for  80  days  under  the  agree- 
ment that  he  was  to  receive  $4  for  each  day  he  worked, 
but  was  to  forfeit  $1.50  for  each  day  he  was  idle.  He 
received  $276  in  all.     How  many  days  was  he  idle? 

4.  A  number  consists  of  two  digits,  the  sum  of  the 
digits  being  10.  If  the  digits  be  reversed  the  new  num- 
ber is  18  larger  than  the  original  number.     What  is  the 

number? 
10— u.  A. 


146  UNIVERSITY   ALGEBRA. 

Let  ^=the  digit  in  tens'  place. 

Since  the  sum  of  the  digits  is  10;  therefore, 

10— ;i;=the  digit  in  units'  place. 
Hence  the  number  expressed  by  these  digits  is 

10;c+10-x. 
But  the  number  expressed  when  the  digits  are  reversed  is 

10(10-^)  +  ^. 
Since  this  is  18  larger  than  the  original  number;  therefore, 

l0(10-^)  +  ;c-(10^+10-^)=:18. 
Removing  parentheses,  100  — 10;c  +  x  — lOx  — 10+x=18. 
Transposing  terms,  —  10x-|-;r— 10^  + x=18  — 100+10. 

Uniting  terms,  —lSx=  —  T2. 

Therefore,  x=4,  one  digit. 

Therefore,  10—^=6,  the  other  digit. 

Hence  the  number  is  46. 

5.  A  number  consists  of  two  digits  of  which  the  sum 
is  12.  If  we  subtract  18  from  the  number  we  obtain  the 
number  with  digits  reversed.     What  is  the  number? 

6.  A  man  is  32  years  old  and  his  son  11.  In  how 
many  years  will  the  father  be  4  times  as  old  as  his  son  ? 

7.  A  man  is  /  years  old  and  his  son  s.  In  how  many 
years  will  the  father  be  n  times  as  old  as  his  son? 

8.  A  mistress  promised  her  servant  $150  a  year  and  a 
new  dress.  The  servant  left  at  the  end  of  10  months 
and  received  $120  and  the  dress.  What  was  the  value  of 
the  dress  ? 

9.  A  train  leaves  a  station  and  travels  at  the  rate  of 
25  miles  an  hour.  Two  hours  later  another  train  follows 
it,  traveling  at  a  rate  of  35  miles  per  hour.  How  long 
before  the  second  train  will  overtake  the  first  ? 

10.  The  body  A  travels  one  yard  a  minute  and  is  pur- 
suing B,  which  is  10  yards  ahead  of  it.  If  A  moves  12 
times  as  fast  as  By  how  many  minutes  before  A  and  B 
are  together? 


SIMPLE    EQUATIONS.  147 

11.  At  what  time  between  2  and  3  o* clock  are  the 
hands  of  a  clock  together  ? 

The  minute  hand  moves  over  one  minute  space  in  one  minute.  At 
2  o'clock  the  minute  hand  is  pursuing  the  hour  hand,  which  is  10 
spaces  ahead  of  it.  The  minute  hand  moves  12  times  as  fast  as  the 
hour  hand. 

12.  Chicago  and  Madison  are  130  miles  apart.  At 
6  p.  M.  a  train  leaves  Chicago  for  Madison  and  runs  at 
the  rate  of  30  miles  an  hour.  A  7  p.  m.  a  train  leaves 
Madison  for  Chicago  and  runs  at  the  rate  of  20  miles  an 
hour.     When  and  where  will  the  two  trains  meet  ? 

13.  A  can  do  a  piece  of  work  in  50  days,  B  can  do  the 
same  work  in  60  days,  and  C  can  do  the  same  work  in  75 
days.  How  many  days  will  it  take  them  to  do  the  work 
together? 

Let  X  represent  the  number  of  days  it  takes  the  three  men  to  do 

the  work  together.     Then  -  is  the  part  of  the  work  they  all  do  in  one 

day.     Since  in  one  day  A  alone  does  ^  of  the  work,  B  alone  does  ^^ 
of  the  work,  and  C  alone  does  ^  of  the  work,  therefore, 
1       1    '  1_1 

14.  A  bath  can  be  filled  by  the  cold  water  pipe  in  10 
minutes  and  by  the  hot  water  pipe  in  15  minutes.  It  can 
be  emptied  by  the  waste  pipe  in  8  minutes.  A  man  hav- 
ing left  all  these  pipes  running,  returns  after  a  time  and 
finds  the  bath  half  full.     How  long  was  the  man  absent  ? 

15.  Suppose  that  in  the  preceding  problem  it  had  been 
stated  that  the  man  returned  after  an  absence  of  16 
minutes.     In  what  condition  would  the  bath  be  found  ? 

16.  Two  barrels,  holding  48  gallons  each,  are  to  be 
filled  with  varnish  worth  70  cents  a  gallon.  The  varnish 
is  to  be  taken  from  two  sorts,  one  worth  60  cents  per 
gallon  and  the  other  worth  90  cents  per  gallon.  How 
much  of  each  sort  is  required  ? 


148  UNIVERSITY   ALGEBRA. 

17.  The  difference  of  the  squares  of  two  consecutive 
numbers  is  19.     What  are  the  numbers? 

18.  A  person  had  $1000,  part  of  which  he  loaned  at  7 
percent,  and  the  rest  at  6  per  cent.;  the  total  interest 
received  was  $63.    How  much  was  loaned  at  7  per  cent.? 

Let  5C=the  number  of  dollars  loaned  at  1%. 

Then  1000— ^=the  number  of  dollars  loaned  at  Q%. 

Therefore,  — —  =:the  interest  received  at  1%. 

and  L-p-r =the  interest  received  at  6%. 

Since  the  whole  interest  received  was  $63,  therefore, 

7%_^6(1000-ic)___ 

100+       100^-^^' 
whence  x=:300. 

19.  A  man  invested  $1100  at  5  per  cent,  interest,  and 
5  years  later  he  invested  $1000  at  7  per  cent,  interest. 
How  many  years  from  this  last  date  until  the  principals 
and  interests  of  the  two  investments  will  equal  each  other? 

20.  A  man  invested  a  dollars  at  n  per  cent.,  and  d 
years  later  he  invested  b  dollars  at  r  per  cent.  How  long 
until  the  two  sums,  principal  and  interest,  will  be  equal  ? 

21.  The  sum  of  $1325  is  borrowed,  to  be  paid  back  in 
two  equal  annual  payments,  allowing  8  per  cent,  simple 
interest.     Find  the  annual  payments. 

22.  A  man  can  row  5  miles  an  hour  down  stream  and 
3  miles  an  hour  up  stream.  If  he  starts  down  stream  at 
1  o'clock  how  far  can  he  go  so  that  he  may  rest  ashore 
an  hour  and  get  home  at  6  o'clock  ? 

23.  A  man  walks  from  A  to  B  and  back  in  a  certain 
time  at  the  rate  of  3|-  miles  an  hour.  When  he  walks  3 
miles  an  hour  to  B  and  4  miles  an  hour  back  it  takes  him 
5  minutes  longer.     Find  the  distance  from  A  to  B. 


SIMPLE    EQUATIONb.  I49 

24.  A  train  leaves  ^4  at  11  a.  m.  for  B,  and  travels  at 
the  rate  of  25  miles  an  hour.  Another  train  leaves  C  at 
noon,  and  runs  through  A  to  B  at  the  rate  of  35  miles  an 
hour,  arriving  at  B  24  minutes  later  than  the  first  train. 
The  distance  from  C  to  A  being  21  miles,  find  the  dis- 
tance from  A  to  B. 

25.  A  passenger  train,  431  feet  long,  going  41  miles  an 
hour,  overtakes  a  freight  train  on  a  parallel  track.  The 
freight  train  is  going  28  miles  an  hour  and  is  731  feet 
long.  How  long  does  the  passenger  train  take  in  pass- 
ing the  other? 

26.  Susie  was  born  in  eighteen  hundred  and  seventy  x. 
Agnes  was  born  the  next  year  and  Mabel  was  born  two 
years  later  than  Agnes.  When  Susie  was  x  years  old 
the  sum  of  the  ages  of  the  three  was  20  years.  Find  the 
year  of  birth  of  each. 

27.  The  metal  of  a  solid  sphere,  radius  r,  is  made  into 
a  hollow  sphere,  internal  radius  r;  required  its  thickness. 

28.  A  cylinder  of  hickory  encloses  a  cylinder  of  iron, 
the  whole  containing  30  cubic  inches  and  weighing  52 
ounces.  If  the  hickory  weighs  .5  ounce  per  cubic  inch 
and  the  iron  4.2  ounces  per  cubic  inch,  how  many  cubic 
inches  of  iron  are  there  in  the  cylinder? 

29.  A  mass  of  tin  and  lead  weighing  180  lbs.  loses  21 
lbs.  when  weighed  in  water,  and  it  is  known  that  37  lbs. 
of  tin  lose  5  lbs.  and  23  lbs.  of  lead  lose  2  lbs.  in  water. 
What  are  the  weights  of  tin  and  lead  in  the  mass  ? 

30.  At  two  stations,  A  and  B,  the  prices  of  coal  are 
$4.50  and  $5.00  per  ton,  respectively.  If  the  distance 
from  A  to  B  is  160  miles  and  the  freight  rate  is  f  cents 
per  ton  per  mile,  find  the  distance  from  A  of  sl  station  at 
which  it  is  immaterial  whether  coal  be  bought  at  A  or  B. 


150  UNIVERSITY    ALGEBRA. 

GKNKRAI.   EQUATION   OF   FIRST   DKGRKE. 

233.  General  Equation.  It  is  evident  that  a;r+^=0 
represents  any  equation  of  the  first  degree  containing  one 
unknown  number ;  for  it  provides  for  any  coefficient  of  x 
and  for  any  term,  or  Absolute  Term  as  it  is  called,  not 
containing  the  unknown  number.  The  root  of  this  equa- 
tion is and  it  is  seen  that  the  equation  may  be  put  in 

the  form,  ^~\ J"^   * 

That  is,  any  equation  of  the  first  degree  may  be  put  in 
the  form  x—theroot=0. 

If  it  is  not  plain  that  every  simple  equation  containing  but  one 
unknown  number  may  be  put  in  the  form  ax-\-b=.0,  the  following 
will  tend  to  make  it  clear.     Take  the  equation 

Ifx  +  l     ■y-3]_[.r+l     x-l 

41    2  6    J~[    8    "^   12 

.  ^+1     ^—3     x+1     x—\ 

Multiplymg  by 4,  -^ —z=:—^Jr—^ 

Multiplying  by  6,     3x  +  3-(^-3)=3x  +  3+2^-2. 

Transposing,  3;ir-x-3%— 2x  +  3  +  3-3  +  2=0. 

Uniting  terms,  -3:?c4-5=0. 

This  is  in  the  form  ax-\-b=Q.     The  root  is  |,  and  the  equation  may 

be  written  x— 1=0. 

234.  Discussion  of  the  Equation.    Since  the  root 

of  the  equation  ax-^-b^^O  is ,  it  follows  that  the  root 

is  negative  if  a  and  b  have  like  signs,  hnt  positive  if  a  and 
b  have  unlike  signs.  The  three  following  cases  are  im- 
portant : 

I.    Suppose  b=0  and  a^O.     In  this  case  the  equation 

becomes  ax^^Q  and  the  root  takes  the  form .     It  is 

.  a 

plain  that  in  this  case  the  value  of  x  is  0. 

The  symbol  =^  stands  for  "not  equal  to,''  and-^t!  for  **not  less  than.*^ 


SIMPLE    EQUATIONS.  151 

II.  Suppose  a=0  a7id  d=0.  In  this  case  the  equa- 
tion becomes  Ox+0=0.  This  equation  can  be  satisfied 
by  any  value  of  x  whatever,  since  0  times  any  number  is 
itself  0.  In  this  case  the  root  takes  the  form  %,  which, 
because  of  the  fact  just  mentioned,  is  often  called  the 
Indeterminate  Form. 

III.  Suppose  a=0  and  b^O.  In  this  case  the  equa- 
tion becomes  Ox-{-b=0  and  the  root  takes  the  form  — -tt. 

If  in  the  equation  ax+b=0  we  put  ^=3-^,  we  get 
x=  —  1^0b ;  if  we  put  ^=tto  o"'  we  get  x=  — 1000^  if  we 
put  ^=To"i7nr'  we  get  jr=— 10000<^.  Thus  we  see  that  in 
the  equation  ax-\-b=0,  as  we  take  a  numerically  smaller 
and  smaller,  the  value  of  x  becomes  numerically  larger 
and  larger.  Evidently,  then,  so  long  as  a  is  not  zero, 
there  is  a  value  of  x  which  will  satisfy  the  equation,  no 
matter  how  near  zero  we  take  a,  but  for  ^=0  there  is  no 
finite  value  of  x  which  will  satisfy  the  equation.  That 
is,  if  ^=?^0  the  equation  Ojtr+^=0  is  an  absurdity  for  finite 
values  of  x, 

GKNKRAI^IZKD   PROBLKMS. 

I.  Two  couriers,  A  and  By  are  traveling  the  same 
road,  the  former  at  the  rate  of  a  miles  per  hour,  the  latter 
at  the  rate  of  b  miles  per  hour.  At  noon  they  are  d  miles 
apart.     When  are  they  together? 

Solution  of  the  Problem.  Let  x  equal  the  number  of  hours  that 
elapse  before  the  couriers  are  together. 

Therefore,         «5C= number  of  miles  A  goes  until  they  meet, 
and  <^x=number  of  miles  B  goes  until  they  meet. 

Since  the  first  must  go  d  miles  more  than  the  second,   therefore, 

whence  ^= 7. 

d  ^~ 

That  is,  they  meet  in 7  hours. 


152  UNIVERSITY   ALGEBRA. 

Discussion  of  the  Problem.  First,  suppose  that  the  couriers  are 
traveling  the  road  in  the  same  direction.  Then  a  and  b  are  of  the 
same  sign. 

d 

I.  \i  ayb,  that  is,  if  A  travels  faster  than  B,  then r,  or  the  value 

a  —  b 

of  X,  is  positive ;  so  that  they  will  meet  at  some  time  after  noon.  It 
will  be  but  a  short  time  after  noon,  (1)  if  ^  is  small,  or  (2)  if  a  is  much 
greater  than  b.  On  the  other  hand,  it  will  be  a  long  interval  after 
noon,  if  (1)  d  is  large,  or  if  (2)  b  is  nearly  equal  to  a. 

II.  \i  a<ib,  that  is,  if  B  travels  faster  than  A,  then is  nesrative- 

a—b  ^ 

so  that  the  couriers  were  together  at  some  time  before  noon.  The 
interval  before  noon  is  small,  (1)  if  d  is  small,  or  (2)  if  b  is  much 
larger  than  a.  The  interval  before  noon  is  large,  (1)  if  d  is  large,  or 
(2)  if  a  is  nearly  equal  to  b. 

III.  If  d—{)  and  ^=?t(5,  that  is,  if  they  were  no  distance  apart  (at 

d 
noon),  then r,  or  the  value  of  ?c,  is  0;  that  is,  the  couriers  are  to- 

a  —  b 

gether  0  hours  from  noon,  or  at  noon. 

IV.  \i  a^b  and  d-=f=.^,  that  is,  if  they  travel  at  the  same  rate,  then 

d  d 

7  takes  the  form  -  and  the  couriers  are  never  together. 

a  —  b  0 

See  Art.  234.  III. 

V.  \i  a—b  and  d—^,  that  is,  if  they  travel  at  the  same  rate  and  are 

d  0 

together  (at  noon),  then y  takes  the  indeterminate  form  -,  and  the 

couriers  are  always  together,  or  any  value  of  x  will  fulfill  the  conditions. 

Second,  if  the  couriers  are  traveling  in  opposite  directions,  then  one 
of  the  following  cases  must  arise  : 

(1)  A^ ^  ^ B 

(2)  ^ A  B ^ 

In  the  first  case  a  is  positive  (if  we  reckon  distance  to  the  right  posi- 
tive) and  b  is  negative.  In  the  second  case  a  is  negative  and  b  is  positive. 

d  d 

I.  If  «  is  +  and  3  is  —  then  - — -  becomes  — --rj  (v^^here  a  and  b '  are 

a^b  a-\-b' 

positive),  which  is  positive.    Hence  the  time  of  meeting  is  after  noon. 

d  d 

II.  If  ^  is  —  and  b  is  -\-  then r  becomes -; — r  (where  a '  and  b 

a  —  b  —a'—b 

are  positive),  which  is  negative.  Therefore  the  time  of  meeting  was 
before  noon. 


SIMPLE    EQUATIONS.  153 

d  d 

III.  If  ^=0  and  «=?^0  and  b^^,  then  both —  and — -  are  0, 

a-\-b  —a'—b 

and  the  couriers  are  together  0  hours  from  noon,  or  at  noon. 

IV.  If  ^=0  and  «=0  and  ^=0,  that  is,  if  they  are  together  (at  noon) 

d  d 

and  do  not  move  at  all,  both  - — -—  and — r  take  the  indeterminate 

a-\-b  —a    —0 

form  g,  and  the  couriers  are  together  for  all  values  of  x,  or  for  all  time. 

2.  A  train  traveling  a  miles  per  hour  is  d  miles  from  a 
train  traveling  b  miles  per  hour  in  the  same  direction. 
When  are  the  trains  together? 

Discuss  the  problem  for:  I,  ayb,  d^O',  II,  a<b,  drf=.^\  III,  d—^, 
a^zb  ;   IV,  a  =  b,  d^{)  ;  .V,  d—Q,  a—b. 

3.  A  cistern  can  be  filled  by  two  pipes  in  a  minutes 
and  b  minutes,  respectively,  and  emptied  by  a  third  in  c 
minutes.  In  what  time  will  it  be  filled  by  all  these  pipes 
running  together? 

abc 

The   result    is   -. >    minutes.       Discuss   the   problem    for : 

bc-\-ac  —  ab  ^ 

X        1  1  1  J      XT       1  1  1 

I,  -<-+t;   and  II,  -=-+T- 
cab  cab 

4.  A  barrel  holding  q  gallons  is  to  be  filled  with  var- 
nish worth  n  cents  per  gallon.  The  varnish  is  to  be 
taken  from  two  sorts,  one  worth  a  cents  per  gallon  and 
the  other  worth  b  cents  per  gallon.  How  much  of  each 
sort  is  required  ? 

,     .     gin  —  b)       ,,                                     ,  q[a  —  n) 
The  result  is 7-^  gallons  at  a  cents  and  — 7-^  gallons  at  b 

a—b  a—b     ^ 

cents  per  gallon.     Discuss  this  result  for:  I,  ayw^b;  II,  n^a^b; 
III,  b'>n>a:  IV,  a>byn\  V,  a—b—n\  VI,  a=n>b. 

5.  Let  s^  and  ^2  ^^  ^^^  specific  gravities  of  two  sub- 
stances of  which  a  compound  is  to  be  made  whose  specific 
gravity  is  6"  and  its  absolute  weight  w.  How  much  of 
each  substance  is  required  ? 

tJAt  O       (  O         ,__      X  \  12) S       (    S    S       \ 

The  result  is    ^^  ^ — -^  of  the  first  and  -^ ^  of  the  second 

substance.     Discuss  the  problem    for:    I,  S=Sj^:^s^;    11,  s^^—s^z^S; 
III,  s^=s^  =  S;  IV,  s^>S>S2;  V.  s^>S2>S. 


154  UNIVERSITY   ALGEBRA. 

6.  At  two  stations,  A  and  B,  the  prices  of  coal  are  $^ 
per  ton  and  %b  per  ton,  respectively.  If  the  distance  be- 
tween A  and  B  is  d  miles  and  the  freight  rate  for  coal  is 
%r  per  ton  per  mile,  find  the  distance  from  A  of  a  station 
on  the  line  at  which  it  is  immaterial  to  a  dealer  whether 
he  buys  coal  at  A  or  at^. 

The  result  is ^ miles  from  A.     Discuss  the  problem  for : 

I,  d-\-rd>a;    II,  d-\-rd<a;    III,  d  +  rd=a;    IV,  dz=za;  V,  r=0,  b^a\ 
VI,  r=rO,  b-a. 

Historical  Note.  The  solution  of  linear  equations  was  known 
to  the  Egyptians.  In  the  British  Museum  is  a  hieratic  papyrus  which 
was  compiled  by  Ahmes,  in  the  reign  of  king  Ra-a-us,  some  time  be- 
fore 1700  B.  C.  This  ancient  Egyptian  mathematical  handbook  con- 
tains, among  other  matter,  eleven  simple  equations  with  one  unknown 
quantity.  The  unknown,  x,  is  called  Hau  (heap).  Equations  appear 
as  follows :  Heap  its  |,  its  \,  its  f ,  its  whole,  it  makes  37;  i.  e. 
\x-\-\x-\'\x-\-xz=Z^,  Early  Greek  mathematicians  were  familiar 
with  only  geometrical  solutions  of  equations,  but  Diophantus  (about  360 
A.  D. )  and  the  Hindoos  were  well  versed  in  solutions  of  linear  equations. 


CHAPTER  XI. 

SIMUIvTANKOUS   EQUATIONS. 

235.  It  is  plain  that  in  the  equation  x+y=25  the 
value  of  either  ^  or  jk  can  easily  be  found  if  the  value  of 
the  other  is  given.  Tlius,  if  j/=l,  ^=24;  ifj/=S,  x—Tl\ 
if  jj/=10,  x=16,  etc.  Thus  it  is  plain  that  there  is  an 
unlimited  number  of  pairs  of  values  of  x  and  y  which 
satisfy  the  equation  x-\-y=2h.  The  same  truth  may 
be  observed  in  any  equation  containing  two  unknown 
numbers. 

236.  Equations  in  which  an  unlimited  number  of 
values  can  be  found  for  the  unknown  numbers  are  called 
Indeterminate  Equations. 

237.  Let  us  consider  the  two  equations 

x+y=2h  (1) 

and  x=^l-\-y  (2) 

in  which  the  x  and  y  are  supposed  to  stand  for  the  same 
numbers  in  one  equation  that  they  stand  for  in  the  other. 

Since  x=^1-\-y,  we  may  substitute  this  value  iox x  in  (1). 
We  then  have  7+jK+j|/=25.  (3) 

We  now  have  a  vSingle  equation  containing  but  one  un- 
known number.  Solving  this,  as  in  the  last  chapter,  we 
find  j/=9, 

and  since  x=^-\-y,  x—\%. 

Notice  that  this  pair  of  values,  7= 9  and  ^=16,  satisfies  both  of  the 
original  equations. 

In  general,  it  will  be  found  that  the  values  of  two  iin- 
knowji  numbers  can  be  found  if  two  equations  containing 
them  are  given. 


156  UNIVERSITY    ALGEBRA. 

238.  Elimination.  If  we  can  obtain  from  two  equa- 
tions containing  two  unknown  numbers,  a  single  condi- 
tional equation  containing  but  one  of  the  unknown 
numbers,  as  above,  the  value  of  this  unknown  number 
can  be  found  in  the  way  explained  in  the  preceding 
chapter.  When  from  two  equations  containing  two  un- 
known numbers  we  obtain  one  equation  containing  but 
one  of  the  unknown  numbers,  we  are  said  to  have  Elim- 
inated the  other  unknown  number. 

239.  Two  or  more  equations  containing  several  un- 
known numbers  so  related  that  they  are  all  satisfied 
simultaneously  by  the  same  set  of  values  of  the  unknown 
numbers  are  called  Simultaneous  Equations,  and  are 
spoken  of  collectively  as  a  Set  or  System. 

We  will  explain  three  methods  of  elimination :  I.  By 
Substitution.     II.    By    Comparison.     III.    By  Addition 

and  Subtraction. 

■* 

KI.IMINATION   BY  SUBSTITUTION. 

240.  We  give  a  few  examples  worked  by  the  method 
of  substitution. 

(1)  Find.;randjKif    'lx=^y—Zd>  (1) 

and  jj/+23=3.r  (2) 

Finding  the  value  oi  x  in  terms  of  y  from  (1),  we  get 

;j;=3r-19.  (3) 

Substituting  ^y—Vd  for  x  in  (2),  we  get 

jj/+23=9y-57  (4) 

Transposing,  j^—9ji/= —57—23.  (5) 

Uniting  terms  and  dividing  both  sides  by  —1, 

8r=80;  (6) 

whence  y=lO,  (7) 


SIMULTANEOUS    EQUATIONS.  I  5/ 

Substituting  this  value  for  y  in  (3),  we  get 

:r=30— 19=11; 
whence  x=\l  and  j^=10. 

To  verify,  we  substitute  these  values  in  the  original 
equations  and  get 

22=60-38  and  10+2^=33. 

(2)  Find  ;r  and  J/ if    1x+^y=im  (1) 

and  3:r-j/=20  (2) 

Finding  the  value  of  y  in  terms  of  x  from  (2),  we  get 

jK=3^-20  (3) 

Substituting  3:r— 20  for  j  in  (1),  we  find 

7-r+ 9^-60=  100.  (4) 

Transposing  and  uniting  terms, 

16:r=160;  -  (5) 

whence  x=10.  (6) 

Substituting  this  value  for  x  in  (3),  we  have 

j^=30-20=10- 
w^hence  ;i:=10  and  y=10. 

241.  It  is  easy  to  see  that  the  method  of  elimination 
used  in  these  examples  may  be  applied  in  the  case  of  any 
set  of  two  simultaneous  equations.     Hence  we  may  say: 

From  either  equatiori  express  one  of  the  unknown  num- 
bers in  terms  of  the  other,  and  substitute  this  value  in  the 
other  equation. 

BI.IMINATION   BY   COMPARISON. 

242.  We  give  a  few  examples  of  elimination  of  an  un- 
known number  by  the  method  of  comparison. 

(1)  Find  ;»;  and  j>/ if    Zx=lZ—y  (1) 

and  2;r=j/+32  (2) 

From  (1),  ^=^^-  (3) 


158  UNIVERSITY    ALGEBRA. 

From  (2),  ^=^^^-  (4) 

Therefore.  ^=-^-  © 

Clearing  of  fractions, 

2(73-j^)=3(^+32).  (6) 

That  is,  146-2:k=3j/+96.  (7) 

Hence,  by  transposing  and  uniting  terms, 

6^=50;  (8) 

whence  j/=10.  (9) 

7Q 10 

Then,  from  (3),  ^=:i£^=21.  (10) 

Hence,  ;i;=21  and  _7=10. 

(2)  Find  ^  and  jj/ if    9jj/+8;r=41  (1) 

and  ll;r-7j/=37  (2) 

From  (1),  ^=^^T^-  (^) 

From  (2),  ^=^^^^-  (4) 

Whence  — g — = j^ (5) 

Clearing  of  fractions, 

7(41-8;»;)=9(ll;t;~37).  (6) 

That  is,  287-56;t:=99.;t;-333.  (7) 

Hence,  by  transposing  and  uniting  terms, 

155;i;=620.  (8) 

Therefore,  x=4. 

n^^        r         /ox  41-32     , 

Then,  from  (3),  j^= — ^ — =1. 

Hence,  x=4:  and  jk=1. 

243.  The  above  is  sufficient  to  show  us  how  to  elim- 
inate an  unknown  number  by  the  method  of  comparison. 

Express  the  same  unknown  number  in  terms  of  the  other 
from  each  equation^  and  equate  the  expressions  thus  fou7id. 


SIMULTANEOUS   EQUATIONS.  I59 

ELIMINATION   BY   ADDITION   AND   SUBTRACTION. 

244.  The  elimination  in  the  following  examples  is 
done  by  the  method  of  addition  and  subtraction. 

(1)  Find  ;tr  and  j>/ if  x+y=b1^  (1) 
and                                      ;»;~jj/=333  (2) 

Addilig  the  left  members  and  the  right  members  of 

these  two  equations,  we  obtain 

^+^,=  579  (3) 

x—y=^Z^  (4) 

2;tr=912  (5) 

whence  x=Ab^  (6) 

Subtracting  the  members  of  (4)  from  the  corresponding 

members  of  (3),  we  get      2j/=246; 

whence  j/=123. 

Therefore,  ;r=456  and  j^=123. 

(2)  Find  ;t:  and  j^  if  15;r:-8r=30  (1) 
and                                       Zx^2y=lb  (2) 

Multiplying  both  members  of  the  second  equation  by 
5,  we  have  for  the  two  equations 

Ibx-  8jj/=3,0  (3) 

\bx-\-10y=lb  (4) 

By  addition,  — 18>/=— 45  (5) 

whence  J^=2^.  (6) 

Hence,  from  (1),  15;tr~20=30.  (7) 

Therefore,  ^=3^  (8) 

Or  the  value  of  x  may  be  obtained  in  another  way. 
Multiplying  both  members  of  the  second  of  the  given 
equations  by  4,  we  have  for  the  two  equations, 

lbx-Sy=ZO  (9) 

12.r-f8^=6Q  (10) 

By  addition,  27^=90  (11) 

whence  ^^fy  or  3-|-. 


l60  UNIVERSITY    ALGEBRA. 

(3)  Find  ;»;  and  J/ if    ll;i;4-12j/=100  (1) 

and  dx-i-   8y=  80  (2) 

Multiplying  both  members  of  the  first  equation  by  9, 
and  both  members  of  the  second  by  11,  we  obtain 

99j^  +  108r=900  (3) 

99^+  88j/=88Q  (4) 

By  subtracting,  20>/=  20  •         (5) 

whence  J^=l.  (6) 

Substituting  this  value  for  y  in  either  of  the  original 
equations,  we  find  x=S, 

245.  The  above  examples  show  us  how  to  eliminate 
an  unknown  number  by  the  method  of  addition  and  sub- 
traction. 

Multiply  both  members  of  the  equations  by  such  numbers 
as  will  make  the  7iumerical  coefficients  of  one  of  the  unknown 
numbers  the  same  in  the  resulting  equations;  then  by  addi- 
tion or  subtraction  we  can  form  an  equation  co?itaining  only 
the  remaining  7ium.ber. 

SPKCIAI.  EXPKDIKNTS. 

246.  The  student  will  find  that  elimination  by  addi- 
tion and  subtraction  is  in  most  cases  the  shortest  method. 
Occasionally,  however,  an  example  will  be  found  which  is 
more  readily  solved  by  one  of  the  other  methods.  Some- 
times, too,  special  expedients  will  still  further  abbreviate 
the  processes.  We  give  a  few  examples  of  this. 

(1)  Solve  23^4  39j/=193,     39;i;+23j/=241. 

By  adding  the  members  of  the  two  equations,  we  get 
62x-{-62y=4S4.  (1) 

By  subtracting  the  members  of  the  second  equation  from 
those  of  the  first,  we  have 


SIMULTANEOUS   EQUATIONS.  l6l 

16;ir-16>/=48.  (2) 

From  (1)  we  get  x-hy=7.  (3) 

From  (2)  we  get  x—j/=S,  (4) 

Whence,  by  addition  and  then  by  subtraction, 
x—6  and  j/=2. 
If  this  example  be  done  in  the  ordinary  way,  large  numbers  will 
be  encountered. 

(2)  Solve  ^--^=1,     -^+-^=6. 

To  solve  these  we  must  first  clear  each  of  fractions, 
giving  9x—  8)/=  12, 

and  14.r+5j/=36, 

which  can  now  be  solved  in  any  of  the  usual  ways. 

,^^  ^  ,      9     4,      18  ,  20     ,^ 

(3)  Solve =1,     — h— =16. 

^  ^  X    y  X      y 

If  these  be  cleared  of  fractions,  the  resulting  equations 
will  involve  the  product  xy^  and  we  would  have  equa- 
tions of  a  kind  not  yet  considered.     But  by  considering 

—  and  —  as  the  unknown  numbers,  we  may  solve  by  the 
x         y 

methods  already  used.  For  example,  by  multiplying  both 

members  of  the  first  equation  by  2,  we  get 

---=2.  (1) 

X     y  ^  ^ 

But  T+V=16-  (2) 

X     y 

28 
By  subtraction,  — =14,  (3) 

2 

whence  -=1  or  j>/=2. 

y 

Therefore,  from  (1),  x=Z, 

We  could  have  eliminated  x  more  readily  if  we  had  first  divided 
both  members  of  the  second  given  equation  by  2. 

11— U.  A. 


l62  UNIVERSITY   ALGEBRA. 

(4)  Solve  i^+l;=2,     1-^=1. 

Multiplying  the  first  equation  through  by  12,  we  have 

^+^=24.  (1) 

X    y 

Muhiplying  the  second  through  by  4,  we  have 

^-^=4.  (2) 

y     X 

Subtracting  (2)  from  (1),     |=20;  (3) 

whence  x=^\  and  y=^. 

EXAMPIyKS. 

Solve  by  any  method : 

I.  3^+8y=19,        2.  2;i:-7j/=8,         3.  19jr-21j/=100 
^x—y=l.  4y—9x=19.  21x—19y=U0 

4.10.+|=210,      8.  1+1=3,  -.f+|=3i 


X 


15     4      .  X    y     1 


10^-^=290.  ---=4.  3-g-2 

X  12  1      1     11 

5.  f +7,-261.  9. -+--10,  .3.j+--g5. 

^■H-«.  -M-F       «-5^4=^' 

2     1_  £,Z_2  ?_^±=?? 

x~j)/~  2"*"  3""  6'  4;r''"5j|/    10* 

7.  x+|=4,  IX.  1+31^=139.  15.  ^^=^x-j, 
24=1.  I  +  -.-31.        1^^=^- 


SIMULTANEOUS   EQUATIONS.  1 63 

16.-—+--         o,  17.  ___+d ^. 

x+j^     x-y_^^  o     x-2y_x    y 

18.  7^-13^=6;tr-10jc-8=0. 
X9.  ^f'-^^=2.-4. 
'  2,-4-^-£=^=3.. 


20. 


~3~       12"""  "60"' 


SIMULTANEOUS   EQUATIONS   CONTAINING  THRKB 
UNKNOWN    NUMBERS. 

247.   If  we  have  two  equations  containing  three  un- 
known numbers,  such  as 

2x+?>y—b2=  9, 

we  can  eliminate  one  of  these  unknown  numbers  by  the 
methods  already  explained,  giving  one  equation  con- 
taining two  unknown  numbers.  Thus,  in  this  particular 
case,  by  multiplying  the  members  of  the  second  equation 
by  2  and  subtracting,  we  find 

7j/-19^=-17. 
Since  an  indefinite  number  of  values  will  satisfy  one 
equation  containing  two  unknown  numbers,  it  follows 
from  this  that  an  indefinite  number  of  sets  of  values  will 
satisfy  two  equations  containing  three  unknown  numbers. 
Suppose,  however,  that  we  have  three  equations  con- 
taining three  unknown  numbers,  as 

2x+?yy-b2=  9,  (1) 

x--2y+l2=l^,  (2) 

Zx-  y-22=  8.  (3) 


164  UNIVERSITY   ALGEBRA. 

Multiplying  both  members  of  (2)  by  2  and  subtracting 
from  (1),  we  get  7y— 19-^=  — 17.  (4) 

Multiplying  both  members  of  (2)  by  3  and  subtracting 
from  (3),  we  get  5y— 23^=-31.  (5) 

Now  we  can  eliminate  y  from  (4)  and  (5)  by  multiplying 
both  members  of  (4)  by  5,  and  both  members  of  (5)  by  7, 
giving  35j/—  95<^=—  85,  (6) 

35>/-161^=-217;  '  (7) 

whence,  by  subtraction,     66<3'=132,  (8) 

w^hence  ^=2. 

Substituting  2  for  z  in  (4), 

7^-38= -17,  (9) 

whence,  j^=3, 

and  substituting  j/=3  and  -3'=  2  in  (1),  we  find  ^=5. 
Therefore,  ^=5,  jv=3,  and  ^=2. 

Here  we  notice  that  we  have  been  able  to  find  the 
values  of  three  unknown  numbers  from  three  equations. 

248.  It  is  evident  that  we  may  proceed  in  a  similar 
way  in  any  case  of  three  linear  equations  containing 
three  unknown  numbers.  That  is,  to  solve  three  simul- 
taneous equations  containing  three  unknown  numbers: 
^  I.  Obtain  from  two  of  the  equations  an  equation  which 
contains  only  two  of  the  unknown  numbers^  by  any  method 
of  elimination, 

II.  From  the  third  given  equation  and  either  of  the 
former  two  obtain  another  equation  which  contains  the  same 
two  unknown  numbers, 

III.  From  the  two  equations  containing  two  unknown 
numbers  thus  found  find  the  values  of  these  unknown  num- 
bers. 

IV.  By  substituting  these  values  in  one  of  the  given 
equations  the  value  of  the  remaining  unknown  number  may 
be  found. 


SIMULTANEOUS   EQUATIONS.  165 

249.  We  further  illustrate  this  subject  by  working  a 
few  examples.  It  should  be  observed  that  while  it  makes 
no  difference  which  one  of  the  unknown  numbers  is  elim- 
inated first,  yet  the  work  is  often  lessened  by  selecting 
for  this  purpose  that  one  of  the  unknown  numbers  whose 
numerical  coefficients  have  the  smallest  I^.C.M. 

(1)  Solve  4x-  by+  ^=  6,  (1) 

1x-ny+2z=  9,  (2) 

x+    j>/+3^=12.  (3) 

The  unknown  number  z  has  the  smallest  numerical 
coefficients,  and  it  is  easier  to  eliminate  it  than  any  of 
the  other  unknown  numbers.  Multiplying  both  mem- 
bers of  (1)  by  2,  we  have 

8jir-10j/+2^=12.  (4) 

Subtracting  (2)  from  this,  gives 

x-\-y=^,     ^  (5) 

Now  multiplying  both  members  of  (1)  by  3,  we  get 

12;i:-15>/+3^=18.  (6) 

Subtracting  (3)  from  this,  we  find 

ll^-16j/=6.  (7) 

We  have  now  to  find  the  values  of  x  and  y  from  (5)  and 
(7).     Multiplying  both  members  of  (5)  by  11,  we  get 

ll;t:-fllj/=33.  (8) 

Subtracting  (7)  from  this,  we  find 

27j/=27;  (9) 

whence  jr=l. 

From  (5),  jr=2, 

and  from  (3),  2  +  1  +  3^=12; 

whence  2'=  3. 

Therefore,  x=2,  y—1,  and  z=Z, 


l66  UNIVERSITY   ALGEBRA. 

(2)  Solve  ^+:k=5,  (1) 

jK+^=7,  (2) 

^+£'=6.  (3) 

This  is  quickly  solved  by  special  expedient.     Thus, 
adding  the  members  of  the  three  equations,  we  get 

2x+2y-^2z=18, 
or  x+y-i-2=d.  (4) 

From  (Ij,  x+y=5;  therefore,  from  (4),  ^=4. 
From  (2),  y-\-z—l\  therefore,  from  (4),  x=2. 
From  (3),  x+z=6;  therefore,  from  (4),  j/=3. 

(3)  Solve  1+^+1=4.  (1) 

i+-l^-^=4.  (3) 

X     y      z  ^  ^ 

Here  we  should  consider  -,  — ,  and  —  as  the  unknown 

X   y  z 

numbers.     Subtracting  (1)  from  twice  (2),  we  get 

^-f^=4.  (4) 

Subtracting  (3)  from  three  times  (2),  we  get 

y      z  ^ 

We  are  now  to  find  —  and  ~  from  (4)  and  (5).    Subtract- 
y         z  ^  ^  ^ 

20 
ing  (4)  from  (5),  T""^' 

whence  <3'=5. 

From  (4),  jj/=4. 

From  (1),  ;tr=3. 


SIMULTANEOUS   EQUATIONS. 


167 


EXAMPLES. 

Solve  the  following  simultaneous  equations: 

1.  ^+j/=37,  3.  2:r+j/=5,  5.  ^t+jK— ^=17, 

y-\-z=22,  by— 2=10.  y-\-2—x=1, 

2.  jj/+-^=14,  4.  5xSy==—l,     6.  5x-\-7y—2z=lS 
^4-:i;=18,  9j/-2^=12,  8;i:+3r+-^=17 
x+y=24:.                Sx+S2=n,            x-4y-i-10z=2S 

II.  ;»;— ^+-s'=5, 


7.  x-{-y+z=SO, 
8x-\-4y+2z=50, 
27x+dy+Sz=64:. 

8.  3.r+2jK+^=23, 
5:r+2j/+4<^=46, 
10^ +5y  4-4.3'=  75. 

9.  4:X-2y+52=lS, 
2x+4:y-S2=22, 
6x+7y—2=6S. 

10.  .;r+J^+4^=23, 
3;i;-j/+22'=ll, 
x+4y—z=4:. 


Sx+4y-5z=lS, 

12.  ;r+2j/+3^=4, 
2;r+3j/+4^=6, 
3;»;+4j/+5^=8. 

13.  ;r+2j/+3^=32, 
jj/+2^+3;i:=40, 
2+2x+^y=4:0. 

14.  ^+3j/+3^=13, 
j/+3^+3;r=15, 
^+3jr+3jF=17. 


2^3 


15.  ^+1=12, 


5     6~   ' 
12+7-^- 


16. 


x+y 

2 


X-\-2 

3 


=  1, 


=  1. 


17-  -+-=1, 
X    y 

y^  z     2 
;»;^0     3 


i9.f+|+J=62. 

3^4^5       '' 

4  +  5+6-^^- 

1,23, 

20.  -+ =1, 

X    y     z 

^+i+^=24, 
.a;    ^     2" 

l-«+-^=14. 

X    y     2 


l68  UNIVERSITY   ALGEBRA. 

LITKRAI.  SIMUWANBOUS  EQUATIONS. 

Solve  the  following: 

I.  x-{-y=2a  and  x—y=23. 

By  addition,  2x=2a  +  2if; 
whence  x=a-^d. 

By  subtraction,  2y=2a--2d; 
whence  y=a—b. 

2.  jr+j/=3«— 2<^,    4.  ax+by=^c,  6.  "Ix—Zy^^hb—a, 

x—y=2a—Bd.         nx+ry=s.  Sx'-2y=a-^5d, 

1,1  n  Jl 

=^.  — — =0.  ax— •€}=—— -' 

X    y  X    y  -^       bd 

8.  ax-\-by=2ab,  9.  a(^a-\- x)^b{b-^y), 

2bx-^2ay=Zb'^—a'^.  ax+2by=d. 

10.   (^-<^):r+(^  +  %=2(a2-^2)^ 
ax^by=a'^-^b'^. 

11.  .r4-:K=2a,  13.  ;ir+jK+^=^,  15.  ax-^by—cz=2ab, 
x+z=2b,  x—y+2=b,  by-]-c2'-ax=2bc, 
y+2=2c.               x+y—2=c.              c2+ax—by=2ac, 

12.  ax-hy=r,       14.  x—nx+y=0,    16.  j»r+J^+<3'=a, 
^— JK=^,  x—ry+2=0,  ny=rx, 
bx-^2=t               x-\-2=t,                    p2=qy, 

17.  x-\-ay^a'^2-\-a^=^^^ 
x-i-by-{-b'^2-^b^=^0, 
x+cy+c^2-}-c^=0, 

o   ^  ,  ^     ^  1  1  a      b     c     ^ 

18.  -+ =;z.      19.  -=a 20.  -+_- f-=3, 

X    y     2  "^   X  y  X    y^  2 

X     y     2       '  y  2  X     y     '2"    ' 

—-  +  -4--=/  1=^—1.  ?^__i_f_o 

;^     7      -3'      *  2  X  X     y     ^""    * 


SIMULTANEOUS   EQUATIONS.  I69 

PROBLEMS. 

1.  The  sum  of  two  numbers  is  70,  and  their  difference 
is  24.     Find  the  numbers. 

Problems  similar  to  this  have  been  worked  with  the  use  of  one  un- 
known number.  We  will  now  work  it  using  two  unknown  numbers. 
Let  t       5C=the  first  number, 

and  let  _>/=the  second  number. 

Then,  because  the  sum  of  the  numbers  is  70,  and  the  difference  of 
the  numbers  is  24;  therefore,     ;r+^=70, 
and  5C— >'=24. 

Solving  these,  we  find  x=47  and  7=33. 

2.  Find  a  fraction  such  that  if  we  add  1  to  the  numer- 
ator, it  becomes  equal  to  ^y  but  if  we  add  2  to  the  denom- 
inator, it  becomes  equal  to  ^. 

Let  5C=the  numerator  of  the  required  fraction, 

and  let  7= the  denominator  of  the  required  fraction. 

Since  the  fraction  with  1  added  to  the  numerator  equals  i,  therefore 

^=-.  (1) 

Since  the  fraction  with  2  added  to  the  denominator  equals  J,  there- 

From  (1)  and  (2)  we  get,  by  clearing  of  fractions, 

2^+2=jjr,  (3) 

3x=y  +  2.  (4) 

Eliminating  y,  we  get  x=4;  whence,  from  (3),  jj^=10.     The  fraction 
is  therefore  ^. 

3.  Find  a  fraction  such  that  when  1  is  added  to  both 
numerator  and  denominator  it  equals  ^,  but  when  3  is 
subtracted  from  numerator  and  denominator  it  equals  ^. 

4.  A  number  is  formed  of  two  digits  of  which  the  dif- 
ference is  3.  If  the  digits  are  reversed,  a  number  is  ob- 
tained which  is  -f-  of  the  original  number.  What  is  the 
original  number? 

Let  X  represent  the  digit  in  tens'  place,  and  y  represent  the  digit 
in  units'  place. 


I/O  UNIVERSITY   ALGEBRA. 

5.  Find  a  fraction  such  that  when  11  is  taken  from 
both  numerator  and  denominator  it  equals  -J-,  but  when  12 
is  taken  from  both  numerator  and  denominator  it  equals  f . 

6.  Two  masons,  A  and  B,  are  building  a  wall,  which 
they  could  finish,  working  together,  in  12  days.  A  works 
8  days  and  B  2  days,  when  the  wall  is  \  done.  How  long 
would  it  have  taken  each  to  have  built  the  wall  ? 

Let  x=the  number  of  days  it  would  take  A, 

and  jj/=the  number  of  days  it  would  take  B, 

In  one  day  A  builds  -  of  the  wall,  and  B  -.     But  together  they  build 

jij  of  the  wall  in  one  day.     Therefore, 

o  o 

In  3  days  A  builds  -  of  the  wall,  and  in  2  days  B  builds  -  of  the  wall. 

"^  X  y 

3    2    1 
Therefore,  from  the  problem,     — \--—-z'  (2) 

^  X    y    ^  ^  ' 

If  we  solve  the  simultaneous  equations  (1)  and  (2),  we  shall  find  the 

values  of  x  and  y. 

7.  A  and  B  can  together  do  a  certain  work  in  80  days; 
at  the  end  of  18  days,  however,  B  is  called  off  and  A 
finishes  it  alone  in  20  days  more.  Find  the  time  in  which 
each  could  do  it  alone. 

8.  A  cistern  holding  4500  gallons  is  filled  by  two 
pipes.  If  the  first  pipe  be  opened  8  minutes  and  the 
second  pipe  1  minute,  400  gallons  will  run  into  the 
cistern ;  but  if  the  first  pipe  be  opened  1  minute  and  the 
second  7  minutes,  600  gallons  will  run  in.  How  much 
water  does  each  pipe  carr}^  in  one  minute?  How  long 
will  it  take  bpth  pipes  to  fill  the  cistern  if  they  are 
opened  together? 

9.  A  and  B  can  do  a  piece  ot  work  in  12  days ;  B  and 
C  can  do  it  in  20  days ;  A  and  C  in  15  days.  How  long 
will  it  take  each  to  do  the  work  alone? 


SIMULTANEOUS    EQUATIONS.  I7I 

10.  A  cistern  can  be  filled  by  two  pipes.  If  both  pipes 
be  opened  for  15  minutes  they  will  fill  ^  ot  the  cistern ; 
but  if  the  first  pipe  be  opened  for  12  minutes  and  the 
second  for  20  minutes,  \  of  the  cistern  will  be  filled. 
How  long  will  it  take  each  of  the  pipes  when  opened 
alone  to  fill  the  cistern? 

11.  A  man  receives  $2160  yearly  interest  on  his  cap- 
ital. If  he  had  loaned  the  same  capital  at  \  per  cent, 
higher  interest  he  would  receive  $240  more  interest 
each  year.  Find  the  amount  of  his  capital  and  the  rate 
per  cent. 

12.  A  cistern  is  filled  with  three  pipes.  The  first  and 
second  will  fill  it  in  72  minutes,  the  second  and  third  in 
120  minutes,  and  the  first  and  third  in  90  minutes.  How 
long  will  it  take  each  of  the  pipes  to  fill  it  ? 

13.  Three  cities,  A,  B,  and  C,  are  at  the  comers  of  a 
triangle.  From  A  through  ^  to  C  is  82  miles:  from  B 
through  C  to  A  is  97  miles ;  from  C  through  A  to  B  is 
89  miles.     How  far  are  the  cities  A,  B,  and  C  apart? 

14.  A  certain  number  consists  of  three  digits,  whose 
sum  is  15.  If  the  first  two  digits  be  reversed  the  number 
becomes  180  larger,  but  if  the  last  two  digits  be  reversed 
the  number  becomes  but  18  larger.  What  is  the  number? 

15.  The  sum  of  three  numbers  is  70.  The  second 
divided  by  the  first  gives  2  for  the  quotient  and  1  for  the 
remainder,  but  the  third  divided  by  the  second  gives  3 
for  the  quotient  and  3  for  the  remainder.  What  are  the 
numbers? 

16.  A  man  rows  30  miles  and  back  in  12  hours.  He 
finds  he  can  row  5  miles  with  the  stream  in  the  same 
time  as  3  against  it.  Find  the  time  he  was  rowing  up 
and  down,  respectively. 


\^2  UNIVERSITY   ALGEBRA. 

17.  In  round  numbers,  it  takes  72  English  and  51  Ger- 
man yards  together  to  make  100  meters.  Also  48  English 
and  84  German  yards  make  100  meters.  How  many  inches 
(English)  in  the  meter?  How  many  inches  (English)  in 
the  German  yard  ? 

18.  A  man  has  two  sums,  one  of  $10000  and  another  of 
$15000,  at  interest,  and  receives  therefrom  $1200  yearly. 
If  the  first  sum  had  been  loaned  at  the  rate  that  the  sec- 
ond bore,  and  if  the  second  sum  had  been  loaned  at  the 
rate  that  the  first  bore,  he  would  receive  $25  less  per 
year.     At  what  rates  were  the  sums  loaned  ? 

19.  A  bicyclist  starts  from  A  to  B  and  back,  and  at 
the  same  time  another  bicyclist  starts  from  B  to  A  and 
back.  They  meet  at  1  o'clock  15  miles  from  A,  and 
again  at  3  o'clock  9  miles  from  A.  What  is  the  distance 
from  A  to  B,  and  what  are  the  rates  of  the  bicyclists  ? 

20.  Two  bodies  move  upon  the  circumferance  of  a  circle 
which  is  100  feet  in  length,  and  meet  every  20  seconds 
when  their  directions  are  the  same  and  every  4  seconds 
when  their  directions  are  opposite.  How  many  feet  does 
each  body  move  per  second  ? 

21.  A  railway  train,  after  traveling  an  hour,  is  de- 
tained 15  minutes,  after  which  it  proceeds  at  f  of  its 
former  rate  and  arrives  24  minutes  late.  Had  the  deten- 
tion taken  place  5  miles  further  on,  the  train  would  have 
been  but  21  minutes  late.  Find  the  original  rate  of  the 
train  and  the  distance  traveled. 

22.  The  specific  gravity  of  lead  is  11.36,  that  of  gutta 
percha  is  .966,  and  that  of  sea  water  is  1.03.  It  is  re- 
quired to  form  of  gutta  percha  and  lead  a  mass  of  5 
pounds  which  shall  have  the  same  weight  as  an  equal 
volume  of  sea  water. 


SIMULTANEOUS   EQUATIONS.  I73 

23.  There  are  two  stations,  A  and  B,  9.1  miles  apart, 
established  for  determining  the  velocity  of  sound.  At 
the  same  instant  a  cannon  is  fired  from  each  station.  At 
A  the  report  of  the  cannon  at  B  is  heard  42  seconds  after 
its  flash  is  seen,  and  at  B  the  report  of  the  cannon  at  A 
is  heard  44  seconds  after  its  flash  is  seen.  Determine  in 
feet  per  second  the  velocity  of  sound  and  the  velocity  of 
the  wind  at  the  time  of  the  experiment. 

GKNKRAI.  SYSTEM   WITH  TWO  UNKNOWN  NUMBERS. 

250.  It  is  evident  that  ax+by^c  represents  any  equa- 
tion of  the  first  degree  containing  two  unknown  numbers, 
for  it  provides  for  any  coefficient  of  x^  any  coefficient 
of  J/,  and  for  any  term  not  containing  an  unknown  num- 
ber. Hence,  a^x+b^y^Cy^,  (1) 
a^x-^b^y^c^y  (2) 
may  be  taken  to  represent  any  system  of  two  equations 
of  the  first  degree  containing  two  unknown  numbers. 

The  symbols  a' .a' ,  a!" ,  etc.,  read  **a prime,*'*  "a  second t''  **a  third,** 
etc.;  or  a^,  a^,  a^,  read  "a  sub  1,"  **a  sub  2,"  **a  sub  3,**  axe  often 
used  for  numbers  which  have  a  common  relation  to  some  other  num- 
ber or  symbol.  Thus,  a^  stands  for  the  coefficient  of  *  in  the  Jlrst 
equation,  and  ^z  j  stands  for  the  coeflQcient  of  x  in  the  second  equation, 
in  this  case  the  suffixes  showing  from  what  equation  the  symbol  is 
taken.  Of  course  such  symbols  stand  for  entirely  different  numbers, 
just  as  though  different  letters  of  the  alphabet  were  used. 

Multiplying  (1)  through  by  a 2  and  (2)  through  hy  a^, 
we  have  a^a^^^x+a^b^y^a^c-^,  (3) 

a^a^x-^-a^b^y^a^c^,  ,     (4) 

Subtracting  (3)  from  (4), 

(a^b.^—a^b^)y=-a^c^—ac^c^\  (5) 

whence  l/^gg^g  W 


1/4  UNIVERSITY   ALGEBRA. 

Again,  multiplying  (1)  by  b^  and  (2)  by  b^,  we  have 

a^b<2,x-^b^b^y=b^c^,  (6) 

a^b^x^-b^b^y^b^c^,  (7) 

Subtracting,  {a^b^—a^b^x—b^c^  —  b^c^y  (8) 

Equations  [1]  and  [2]  may  be  used  as  formulas  for 
solving  any  system  of  simple  equations  containing  two 
unknown  numbers. 

251.  Since  the  denominators  in  the  right  members  of 
[1]  and  [2]  are  the  same,  we  may  write 

X         __         y        1 

^2^1  ~"  ^1^2     ^1^2  ""^2^1     a-^b^—a^bi 

252.  Discussion  of  the  Systems.  From  [1]  and  [2] 
we  observe  the  three  following  cases : 

I.  Suppose  a -^c 2— a 2C  1=0  and  aib2^ci2bi^0;  that  is^ 

suppose  — ^=^=7^=-^.     In  this  case  the  value  oi  y  is  0. 
^2     ^2     ^2 
What  will  result  if  ^2^i~"^i^2=0  and  a-^b^—a^b-^^^O} 

II.  Suppose  ^1^2 — ^2^1=0  ^^^  <3^i^2""<^2^i=0;  that  is, 

suppose  ~=~=T^*     I^  ^^is  case  the  values  of  x  and  y 

^2     ^2     ^2  0  a       c      b 

each  take  the  indeterminate  form  7^.     Now  if  — =-^=-^ 

U  ^2     ^2     ^2 

say=r,  then  aiX+b-i^y=^c-i  can  be  made  from  a2:^^+^2j^ 
=^2  by  multiplying  both  members  by  a  certain  number  r. 
Thus,  one  equation  is  merely  a  repetition  of  the  other 
equation,  and  since  one  of  these  equations  is  satisfied 
by  an  indefinite  number  of  values  of  x  and  j/,  it  follows 
that  an  indefinite  number  of  values  of  x  and  y  satisfy  the 
system  a-^^x+b^y^c^  and  <22jr+<^2j^'=^2- 


SIMULTANEOUS   EQUATIONS.  1 75 

When  the  values  of  x  and  y  take  the  form  %  the  equations 
a'^x-\-b^y=c^  and  a^x-^-b^^y^c^  are  said  to  be  Dependent 
and  the  system  is  satisfied  by  an  iridefinite  number  of  values 
of  X  a?id  y. 

The  student  may  try  to  solve  the  system  3:x;+3>'=6,  8^+8y=16. 

III.  Suppose  a -^c 2^(1 2C  1=^0  and  a -^b 2 — a^bi^O)  that  is, 

suppose  —=7^=7^—.     In  this  case  the  denominators  in 

^2     ^2     ^2 
the  values  of  x  and  y  become  0.     It  is  plain  that  by 
dividing  the  members  of  each  equation  of  the  system  by 
the  coefficient  of  x  the  system  reduces  to 

x+ky=l, 

x+ky=m^ 
where  l^m.    It  is  now  evident  that  one  equation  of  this 
system  contradicts  the  other. 

A         B 
When  the  values  of  x  and  y  take  the  form  -^  and  -^  the 

equations  a^x+b-^y^^c^  and  a2X-\-b2y=C2  are  said  to  be 
Incompatible,  and  the  system  is  satisfied  by  no  finite 
values  of  x  and  y. 

The  student  may  try  to  solve  the  system  2x  +  5y-=9,   4;c  +  10/ -15; 
i.  e.t  the  system  x-k-\y=%,  x+fjK=^. 


CHAPTER  XII. 

QUADRATIC   EQUATIONS. 

253.  In  Art.  21  was  given  a  definition  of  the  degree  of 
a  polynomial  with  respect  to  any  letter  or  letters.  The 
Degree  of  an  Equation  is  its  degree  with  respect  to  the 
unknown  numbers;  i,  e, ,  it  is  the  degree  of  that  one  of  its 
terms  whose  degree  with  respect  to  the  unknown  number 
is  highest. 

Remember  that  the  last  letters  of  the  alphabet  are  used  to  stand 
for  unknown  numbers. 

254.  A  Quadratic  Equation  is  an  equation  of  the 
second  degree.  In  this  chapter  we  deal  only  with  equa- 
tions of  one  unknown  number.  Quadratic  equations  are 
divided  into  two  classes:  Pure  or  Incomplete,  and 
Affected  or  Complete. 

A  pure  or  incomplete  quadratic  equation  is  one  which 
contains  the  second  but  not  the  first  power  of  the  un- 

known  number,  as  3jf2  =  12  and  — = — =2. 

o 

An  affected  or  complete  quadratic  equation  is  one  which 
contains  both  the  second  and  first  powers  of  the  unknown 

x^     X 
number,  as  Zx'^-\'^=Z^  and  —+-=3. 

255.  A  Root  of  an  equation  is  any  number  which 
substituted  for  the  unknown  number  will  satisfy  the 
equation,  i.  ^.,  will  cause  the  equation  to  be  true.  For 
example,  2  is  the  root  of  the  equation  x^+x=6y  for  if  2 
be  written  in  place  of  x  we  get  2^+2=6,  or  4-|-2=6, 
which  is  true.     Again,  —3  is  also  a  root  of  x^+x=6y 


QUADRATIC    EQUATIONS.  1/7 

for  if —  3  be  written  in  place  of  ;i;we  get  (—3)2  — 3=6,  or 
9—3=6,  which  is  true. 

Although  we  deal  in  this  chapter  only  with  equations 
of  the  second  degree,  still  this  definition  of  root  will  hold 
good  for  an  equation  of  any  degree  whatever,  but  it  must 
be  understood  that  the  word  can  be  used  only  with  ref- 
erence to  an  equation  of  one  unknown  number. 

The  student  must  not  confuse  the  root  of  an  equation  with  the  root 
of  an  expression.     See  Art.  190. 

256.  The  Solution  of  an  equation  is  the  process  by 
which  the  roots  are  found. 

PURK   QUADRATIC   EQUATIONS. 

257.  If  -^^=4  we  know  that  x  must  be  some  number 
which  raised  to  the  second  power  will  give  4.  Now 
there  are  two  such  numbers,  2  and  —2;  therefore  x=^ 
or  —2.  If  JT^  be  placed  equal  to  some  other  number, 
there  will  be  two  values  of  x  of  opposite  signs  but  other- 
wise just  alike. 

Solve  the  equation  2  (^r^- 3)— 22=4. 

Performing  indicated  operation, 

2x2 -6-22^:4. 
Transposing,  2;*:2 =4+6+22. 

Uniting  terms,  2x2=32. 

Dividing  by  2,  5C«=16. 

Hence,  ^=4  or  —4. 

258.  To  solve  a  pure  quadratic  equation,  collect  all  the 
terfns  containing  the  unknown  number  in  the  first  member 
of  the  equation  and  all  the  known  terms  hi  the  second  mem- 
ber; unite  each  group  of  terms  into  a  single  term;  divide  by 
the  coefiiciejit  of  the  square  of  the  unknown  number,  and 
extract  the  square  root  of  each  side  of  the  resulting  equation. 


178  UNIVERSITY   ALGEBRA. 

259.  In  solving  a  pure  quadratic  equation  we  usually 
write  both  values  of  the  unknown  number  at  once.  For 
example,  in  the  equation  x^=4j  after  extracting  the 
square  root  of  each  side,  we  write  :r=±2,  using  the 
double  sign  to  show  that  2  or  --2  will  satisfy  the  given 
equation  x^  =  4:. 

The  student  may  think  that  we  should  write  the  double  sign  on 
do^/z  sides  of  the  equation  instead  of  one  side  only,  thus:  ±^=±2. 
But  this  we  know  means  x=2  or  x=—2  or  —x=2  or  —x=  —  2.  The 
third  of  these  equations  is  really  the  same  as  the  second,  and  the 
fourth  is  really  the  same  as  the  first,  so  that  we  really  get  no  more 
values  by  writing  ±x—±2  than  we  do  by  writing  x=#  ±2. 

260.  Whenever  we  extract  the  square  root  of  each 
side  of  the  equation  we  should  write  the  double  sign  ± 
on  one  side  of  the  equation  obtained. 

261.  When  in  treating  a  pure  quadratic  equation  by 
the  method  described  in  Art.  258  we  arrive  at  an  equa- 
tion in  which  the  right-hand  member  is  not  a  perfect 
square,  we  cannot  find  the  square  root  exactly,  but  can 
do  so  approximately.  In  such  cases  we  merely  indicate 
the  square  root;  for  example,  if  we  have  the  equation 
x^  =  S  we  write  ;r=±l/3. 

KXAMPi^BS. 

Solve  the  following  equations : 

1.  x^+S=4.  5.  (:r2  +  l)  +  (^2  +  2)-f  (;r2  +  3)  =  306. 

2.  x^-hS=7.  6.  S(ix^'-l)+4(x'^-2)=5x'^  +  S9. 

3.  3:^2-4=204-^^  7.  (;r2-4)-f  (jr2-f  2)=^2+4^ 

4.  5x''-7=2dS  +  2x\  8.  2(;r2  +  i)^3(^2  +  3)_5^^ 

9.  (x''-l)-(x^-2)-(x^-S')=SiSx\ 


QUADRATIC    EQUATIONS.  1 79 

"•^H3+5  =  ^-  ^3.-y s 10=0. 

12.  2;»;H =Sx 14.  -^ — -+-=-. 

XX  X  x^—1     4     2 

15.  2 (:v2_i)_3 ^^2 4.1)^10=0. 

16.  9;t;2_ie==o.  18.  (2^+l)2=4;f+2. 

17.  12;t2_75=o.  19.  8^2_i99_(;^+l>)2_2;i;. 

SOI^UTION   BY   FACTORING. 

262.  The  equation  ;r^  =  4  may  be  written  in  the  form 
^2—4=0.  Now,  as  the  first  member  of  this  equation  is 
the  difierence  of  two  squares,  it  may  be  factored,  and 
hence  the  equation  may  be  written  (;r— 2)(^+2)=0. 

263.  The  product  of  two  or  more  factors  is  equal  to 
zero  whenever  any  one  of  the  factors  is  zero.  Therefore 
the  equation  (x—2)(ix-{-2)=0  may  be  satisfied  in  either 
of  two  ways:  first,  when  jt— 2=0,  z.  e.,  when  x=2;  and 
second,  when  x-\-2=0,  z.  e.,  when  x==—2. 

As  another  example  take  the  equation 
5;r2 -9=2:^2 +  18. 
Transposing  all  the  terms  to  the  first  member,  we  get 

5^2__2;r2-9-18=0, 
or  8;»;2__27=0. 

Dividing  by  3,  x'^  —9=0. 

Factoring,  (ji;— 3)(^-f3)  =  0. 

This  equation  can  be  satisfied  in  either  of  two  ways : 
first,  when  x—S=0,  z.  e.,  when  x=S;  and  second,  when 
.^+3=0,  z.  e.y  when  x=—S.  » 

264.  By  the  method  illustrated  in  these  two  examples 
we  get  the  same  roots  as  would  be  obtained  by  the  former 
method.  Thus  we  have  another  method  of  solving  a  pure 


l80  UNIVERSITY   ALGEBRA. 

quadratic  equation,  viz. :  Collect  into  one  term  all  the  un- 
known numbers^  and  into  another  term  all  the  known  num- 
bers; write  these  two  terms  on  the  same  side  of  the  equation, 
making  the  other  side  zero;  divide  both  members  by  the 
coefficient  of  the  square  of  the  unknown  number  (remem- 
bering that  when  zero  is  divided  by  any  nu7nber  the  quotient 
is  still  zero);  factor  the  resulting  first  member;  put  each 
factor  separately  equal  to  zero,  and  solve  the  resulting 
simple  equations, 

KXAMPIvBS. 

Solve  the  following  equations  by  the  method  just 
explained : 

1.  .^2-100=0.  6.   (^+2)2=4(:r+5). 

2.  4^2__ioo=0.  7.   (;r+2)(:r+3)=5^+42. 

3.  (2;i;+l)2=4;t:+82.  8.  j^2_^.r+l=;t:+101. 

4.  5;i;2=80.  9.  x2-2:r--3=33-2;r. 

5.  ;i;2  +  i==26.         10.  (;c+^+^)2  =  2(«  +  ^);i;+2(«2+^2)^ 

AFFBCTKD   QUADRATICS. 

265.  If  we  have  given  the  equation  ^2  =  25  we  solve 
it  by  one  of  the  preceding  methods,  and  find  x=±:5. 
Similarly,  if  we  have  the  equation  (^+1)^  =  25  we  find 
x+l=±B.  If  we  take  the  upper  sign  we  get  ^+1=5, 
or^=4,  and  if  we  take  the  lower  sign  we  get  x-{-l=—5, 
or  x=^—6.  Thus  we  find  the  roots  of  the  equation 
(;i:+l)2=25,  or  what  is  the  same,  x^-{-2x+l=25,  or 
x^-\-2x=-24:/ 

266.  Therefore,  to  solve  x^+2x=24y  we  first  add  1  to 
each  member  to  make  the  first  member  a  perfect  square, 
and  get  x'^-\-2x+l  =  2b\  then  we  take  the  square  root  of 
each  member,  and  get  :r4-l=±5,  whence  x=4:  or  —6. 


QUADRATIC    EQUATIONS.  l8l 

Similarly,  to  solve  x'^-\-6x=7,  we  add  to  each  member 
a  number  that  will  make  the  first  member  a  perfect 
square.  Plainly,  9  is  the  number;  hence  x'^  +  6x+9=16. 
Next  we  take  the  square  root  of  each  member,  and  get 
;r+3==b4,  whence  x=l  or— 7. 

267.  Similarly,  to  solve  any  affected  quadratic  equa- 
tion, we  first  reduce  the  equation  to  a  form  in  which  the 
terms  containing  x^  and  x  are  in  the  first  member,  and 
the  term  not  containing  x  is  in  the  second  member;  sec- 
ond, if  the  coefiicient  of  x^  is  not  unity,  we  divide  each 
member  of  the  equation  by  that  coefficient,  so  that  the 
coefficient  of  ^^  shall  be  unity;  third,  we  add  to  each 
member  of  the  equation  a  number  that  will  make  the 
first  member  a  perfect  square,  and  then  take  the  square 
root  of  each  member  and  solve  the  resulting  simple 
equations. 

268.  Adding  to  a  given  expression  a  number  that 
will  make  the  sum  a  perfect  square  is  called  Complet- 
ing the  Square. 

269.  When  the  coefficient  of  jt^  is  unity  what  number 
is  it  that  we  must  add  to  each  member  of  an  equation  to 
make  the  first  member  a  perfect  square?  To  answer  this 
let  us  see  how  a  perfect  square  is  produced.  We  know 
that  (ix+ay^x'^-\-2ax+a^, 

and  (x — a)^  =  x'^ — 2ax+a^. 

Notice  here  that  whatever  number  is  represented  by  a, 
the  third  term  is  the  square  of  one-half  the  coefficient 
of  x;  hence  the  number  to  be  added  to  each  member  of 
the  given  equation  is  the  square  of  one-half  the  coeffi- 
cient of  X. 


1 82  UNIVERSITY   ALGEBRA. 

270.  Hence,  to  solve  any  affected  quadratic  equation: 

I.  Reduce  to  a  form  in  which  both  x^  and  x  are  in  the 
first  member  and  all  terms  not  contai7iing  x  are  in  the  sec- 
ond member, 

II.  If  the  coefficie7it  of  x'^  is  not  already  unity,  divide 
each  member  of  the  equation  by  that  coefficient^  thus  making 
the  coefficient  of  x'^  unity. 

III.  Complete  the  square  by  adding  to  each  member  the 
square  of  one- half  the  coefficient  of  x. 

IV.  Extract  the  square  root  of  each  member  of  the  equa- 
tion and  solve  the  resulting  simple  equations. 

KXAMPivKS. 

Solve  the  following  equations : 

1.  .a:2+4x=5.  7.  ^x'^ —^x-=%,  13.  x'^  —  Vdx-r-  —9. 

2.  ^24.6;r=16.        8.  ;r2-7.r=-6.  14.  1x''-\hx=m. 

3.  2;i;2— 20;r=48.    9.  x'^—ax=^a'^.  15.  x'^-\-^x=  —15. 

4.  ;tr2+3;c=18.      ID.  ^2_2a^=3«2.  16.  3;i:2_|.i2;r=36. 

5.  ;r2+5.r=36.      \i,x''-x=^X  17.  2.r2  +  10x=100 

6.  3;r2  +  6.a:=9.      12.  ^2 +^^^^2+^^  18.  ^2_5^^_4 

19.  3;r2  — 12a.a;=63^^       20.  ^x'^  —  Vlax=\^a'^, 

271.  The  above  method  will  enable  us  to  solve  any 
affected  quadratic  equation  that  may  be  given,  but  fre- 
quently it  will  oblige  us  to  use  fractions,  and  unless  the 
terms  of  the  fractions  are  small  numbers  it  will  be  easier 
to  complete  the  square  by  another  method,  which  we 
will  now  consider. 

272.  We  know  that 

{ax-{'by=^a'^x'^-\-1abx-\rb'^, 
and  lax—by  =  a'^x'^—2abx-^b'^, 

so  that  each  of  these  two  second  members  is  a  perfect 


QUADRATIC    EQUATIONS.  I  83 

square.  We  therefore  seek  to  reduce  the  given  equation 
so  that  the  first  member  shall  be  in  the  form  of  one  of 
these  two  second  members. 

273.  Notice  two  things:  first,  that  the  coefiicient  of 
;i:2  is  a  perfect  square,  and  second,  that  the  third  term 
equals  the  square  of  the  quotient  obtained  by  dividing 
the  second  term  by  twice  the  square  root  of  the  first 
term.  Therefore,  to  reduce  the  first  member  of  any  given 
quadratic  equation  to  either  of  the  forms  a'^x'^+2adx-i-d^ 
or  a'^x^  —  2adx+d'^, 

I.  Reduce  the  equation  to  a  form  in  which  the  terms  con- 
taining x'^  and  X  are  in  the  first  member^  and  all  terms  not 
containing  x  are  in  the  second  member. 

II.  Multiply  each  member  of  the  equation  by  a  number 
that  will  make  the  coefficieiit  of  x'^  a  perfect  square. 

III.  Add  to  each  member  the  square  of  the  quotient 
obtained  by  dividing  the  second  term  by  twice  the  square 
root  of  the  first  term. 

The  rest  of  the  process  of  solution  is  like  that  already 
given,  viz.:  take  the  square  root  of  each  member  and  solve 
the  resulting  si77iple  equations. 

Let  us  solve  by  this  method  the  equation 
3;c^  +  7^  + 11=2^+33. 
Transposing  %x  and  11,  we  get 

3^2 +  5;*:=:  22. 
Multiply  each    member   by  3   or  12   or   27  or  3  times  any  square 
number  and  the  coefficient  of  5C*  will  be  a  perfect  square.     Taking 
the  first  of  these  multipliers,  we  get 

9^2^15x=66. 
Adding  to  each  member  {^Y,  or  (f  )2,  we  get 

Taking  the  square  root  of  each  member,  we  get 

3x+f=±Y-. 
Hence,  3x=:6  or  —11  and  x  =  2  or  —  V** 


I  84  UNIVERSITY   ALGEBRA. 

If  we  had  multiplied  by  12  instead  of  3  we  would  have  obtained 
36:jc2  +  60;^H- 25=264+25=3289. 
Hence,  6?cH-5=±17. 

Hence,  6;c=12  or  -22,  and  x=2  or  —  V- 

kxampi.es. 

Solve  tlie  following  equations: 

1.  Sx^+4x=7.      5.  2x''-S5=Sx.  g.  2;i;2  +  i0:r=300. 

2.  Sx^+6x=24:.     6.  3.^2— 60=5.r.  lo.  3;t:2 —  10^^=200. 

3.  4.x'^—5x=26.     7.  3.^2—24=6^^.  ii.  4x'^—7x-i-^=0. 
/^,  5x^-7 x==24,     S.2x^-Sx=104:.  12.  |:r2-|:i;=-if 

13.  9x^  +  ex-AS=0.  17.  2.^2 -22;^;=- 60. 

14.  18;i;2_3^_6e:=0.        18.  3jr2  +  7;i;-370=0. 

15.  ^x^-Sx+{i=0.  19.  5^2_l^__7__0. 

I<ITKRAI,  QUADRATIC    EQUATIONS. 

274.  Literal  quadratic  equations  may  be  solved  the 
same  as  numerical  ones.  But,  after  the  square  is  com- 
pleted, the  second  member  will  be  a  literal  instead  of  a 
numerical  expression,  and  hence  the  square  root  usually 
cannot  be  taken,  and  so  will  have  to  be  indicated.  Thus, 
to  solve  ;ir2+4<2j<;+^=0  we  proceed  as  follows: 

Transposing  <^,  x'^-\-^ax^:z — b. 

Adding  4^^  to  each  member, 

Taking  the  square  root  of  each  member,    

x  +  2g=±V4^^-/^. 

Transposing  2«,  x——%a±*^^a'^  —  b.  

One  value  of  x  is  —Za-\-x\a^  —  b  and  the  other  value  is  —%a—H^d^—b 


QUADRATIC    EQUATIONS.  1 8$ 

EXAMPLES . 
Solve  the  following  equations : 

1.  x'^-\-1ax-=^b,        3.  x'^—hax=^1b.      5.  1x'^-\-Zax==U, 

2.  x'^-\-4ax=d,        4.  ax'^  —  dx—c.         6.  ax^+a^x=a^, 

7.  x^—6ax—Sd=0.  9.  5;»;2_7^^_|_^^0. 

S.  2x^--6ax—4d=0,       10.  ax^-}-dx+c=^0. 

SOI.UTION   BY   FACTORING. 

275.  When  all  the  terms  of  an  affected  quadratic 
equation  are  transposed  to  the  left  member,  making  the 
right  member  zero,  the  equation  may  be  solved  by  fac- 
toring if  the  resulting  first  member  can  readily  be  ex- 
pressed as  the  product  of  two  factors  each  of  the  first 
degree  with  respect  to  the  unknown  number.  Thus,  to 
solve  the  equation  x^—6x=—6  we  transpose  —6,  and 
obtain  x^—5x+6=0.  By  the  method  of  Art.  128  we 
find  the  factors  of  the  left  member  to  be  x—2  and  x—S, 
Therefore  the  equation  may  be  written  (x—2')(x—S)=0, 
which  is  satisfied  if  x=2  or  if  :r=3. 

This  method  of  solving  an  affected  quadratic  equation 
cannot  be  used  to  advantage  unless  the  left  member  is 
easily  factored,  but  when  easily  factored  this  is  prob- 
ably the  easiest  wa}'-  to  solve  them. 

KXAMPI^KS. 

Solve  by  factoring  the  following  equations : 
t.  x^—x=6.  s.x^+x=12.  g.  ^2_^5^_l_g^0. 

2.  x'^-{-7x=  —  12.  6.  x'^=6x—5.  10.  x^  +  nx=—SO 
S.x'^—5x=U.  7.  x^  =  -'4x+21.  II.  ^2_7^_f_12=o 
4.  ^2+^=12.  8.  x^  =  —4x+5.      12.  ;r2  — 13;t:=30. 


1 86  UNIVERSITY   ALGEBRA. 

13.  x^+2x+l  =  6x+6,  18.  x^  —  (ia-hd')x-\-ad=0. 

14.  x^-A9=10(x-7),  19.  x''-(a-\-l)x+a=0. 

15.  2:^2 +  60;i:= -400.  20.  ^2  +  (^  +  ^)^+^^=,0^ 

16.  10;tr2+600jt;=-8000.  21.  ^2  4.(^  +  i)^4.^==0. 

17.  2;i;2__40;t:=4j^— 240.  22.  x^  +  (ad+l')x+ad=0. 

PROBI.KMS   LEADING  TO   QUADRATIC   EQUATIONS. 

276.  To  solve  a  problem  the  first  thing  to  do  is  to 
form  an  equation,  the  result  of  solving  which  will  fulfill 
all  the  requirements  of  the  problem.  The  formation  of 
this  equation  is  sometimes  a  great  difficulty  to  students, 
but  after  the  equation  is  once  written  down  there  is 
usually  little  or  no  difficulty  with  a  problem.  The  diffi- 
culty here  spoken  of  may  be  largely  overcome  if  the 
student  will  keep  in  mind  the  fact  that  the  equation  is 
formed  by  using  the  unknown  number  in  exactly  the 
same  way  that  any  assumed  result  would  be  used  to  see 
whether  or  not  this  assumed  result  were  right.  This  is 
illustrated  in  the  following  problem. 

A  train  travels  600  miles  at  a  uniform  rate  of  speed. 
If  the  rate  had  been  10  miles  more  an  hour  the  journey 
would  have  taken  5  hours  less.  Find  the  rate  of  the 
train. 

Let  us  see  if  40  miles  an  hour  is  the  rate.  If  the  train  goes  40  miles 
an  hour,  to  go  600  miles  will  require  \%^  or  15  hours.  And  if  the 
rate  were  10  miles  more  an  hour  it  would  be  50  miles  an  hour,  and  to 
travel  600  miles  at  this  rate  would  require  ^^^  or  12  hours.  By  the 
first  supposition  (40  miles  an  hour)  it  requires  15  hours,  and  by  the 
second  supposition  (50  miles  an  hour)  it  requires  12  hours.  As  this 
last  result  (12  hours)  is  not  5  hours  less  than  the  first  result  (15  hours) 
we  conclude  that  the  rate  is  not  40  miles  an  hour. 

Now  let  us  see  if  30  miles  an  hour  is  the  rate.  To  travel  600  miles 
at  30  miles  an  hour  requires  ^^^  or  20  hours,  and  to  travel  600  miles 


QUADRATIC    EQUATIONS.  1 8/ 

at  40  miles  an  hour  requires  ^  or  15  hours;  and  as  20  —  15—5  [z.  e., 
the  second  time  is  5  hours  less  than  the  first),  we  conclude  that  the 
rate  is  30  miles  an  hour. 

Now,  to  form  an  equation  we  say,  let  x  represent  the  number  of 
miles  an  hour  the  train  travels;  then  to  travel  600  miles  at  x  miles  an 

hour  requires  hours,  and  to  travel  600  miles  at  x-^10  miles  an 

600    ^  J  x.      .        .      t- 

hour  requires 77-  hours,  and  as  the  time  m  the  second  instance  is 

^  :>c-flO 

5  hours  less  than  in  the  first  instance,  we  have 

600 600__ 

X       :r+10~ 
From  this  equation  we  have 

5^2  -}-50x=600  (^+10)  -  600^, 
or  5xZ-\-o0x=m00, 

or  ^24-10^=1200. 

Completing  the  square,  we  have 

%2  +  10x+25=1225. 
Extracting  the  square  root,  we  have 

x+5=±35. 
Hence  x==30  or -40. 

The  first  of  these  results  (30)  agrees  with  the  conclusion  reached 
above;  but  here  another  question  arises,  what  is  to  be  done  with  the 
result  —  40  ?  So  far  as  the  algebraic  work  goes  —40  is  as  good  a  result 
as  30,  but  a  train  traveling  —40  miles  an  hour  is  something  void  of 
meaning,  so  this  result  is  rejected  and  30  is  retained  as  the  true  result. 

277.  It  will  often  happen,  as  in  the  example  just 
worked,  that  the  solution  of  the  equation  formed  as 
above  described  leads  to  a  result  which  does  not  apply- 
to  the  problem  we  are  solving.  The  reason  of  this  is 
that  the  algebraic  statement  of  the  problem  (by  means  of 
the  equation  formed  as  above  described)  is  more  general 
than  the  statement  in  words.  It  will,  however,  usually 
be  quite  easy  to  select  the  result  which  belongs  to  the 
problem  we  are  solving,  and  then  we  reject  the  other 
result  as  inapplicable. 


1 88  UNIVERSITY    ALGEBRA. 

PROBLEMS. 

I.  A  cistern  can  be  filled  by  two  pipes  in  33-|-  minutes. 
If  the  smaller  pipe  takes  15  minutes  more  than  the 
larger  one  to  fill  the  cistern,  in  what  time  will  it  be  filled 
by  each  pipe  singly? 

Let  us  see  if  the  smaller  pipe  will  fill  the  cistern  in  45  minutes. 
If  so,  the  larger  pipe  will  fill  the  cistern  in  30  minutes.  If  larger 
pipe  will  fill  the  cistern  in  30  minutes  it  will  fill  ^  of  the  cistern  in 
Icminute,  and  if  smaller  pipe  will  fill  the  cistern  in  45  minutes  it 
will  fill  :^  of  the  cistern  in  1  minute.  Therefore  together  they 
will  fill  s^-faV=tV  of  cistern  in  1  minute.     But   together   they  fill 

1        3 
oqi^^Tofj  of  cistern  in  1  minute,  and  as  ^  is  not  equal  to  y§o  we  con- 
clude that  45  minutes  is  noi  the  time  in  which  the  smaller  pipe  will 
fill  the  cistern. 

Let  ^= number  of  minutes  required  to  fill  the  cistern  by  the  smaller 
pipe;  then  ^  — 15=number  of  minutes  required  to  fill  the  cistern  by 
the  larger  pipe.     If  larger  pipe  will  fill  the  cistern  in  %  — 15  minutes, 

it    will    fill    r-=  of  cistern  in  1  minute;  and  if  smaller  pipe  will 

x-16  -^  ^^ 

fill  the  cistern  in  x  minutes,  it  will  fill  -  of  the  cistern  in  1  minute. 

^11         2x 15 

Therefore  the  two  pipes   together  will   fill  — -j —  or  — -—  of 

I  x  —  15  X  x{x  —  \^) 
cistern  in  1  minute.     Together  they  fill  ^— -  or  j-g^  of  the  cistern  in  1 

minute.     Therefore,  we  have  the  equation 

3        2x-15 


100"^(5C-15) 
Clearing  of  fractions,  3^^  -45:>c=200x -  1500. 

Transposing  200;c,  3^^  _  2455c  =  - 1500. 

Multiplying  by  12,  36x2-2940;*:=  - 18000. 

Adding  (245)^  to  each  member  to  complete  the  square, 

36x2  -29405C+60025=42025. 
Extracting  the  square  root  of  each  member, 

6x- 245  =±205. 
Hence.  6x=450  or  40;  therefore,  x=75  or  6f . 

From  this  it  appears  that  the  smaller  pipe  will  fill  the  cistern  in 
either  75  or  6|  minutes;  but  evidently  6|  is  not  admissible,  for  it  takes 
the  smaller  pipe  15  minutes  more  to  fill  the  cistern  than  it  takes  the 
larger  pipe;  but  it  takes  the  larger  pipe  some  time  to  fill  the  cistern. 


QUADRATIC    EQUATIONS.  1 89 

So  it  is  plain  that  it  must  take  the  smaller  pipe  more  than  15  minutes 
to  fill  the  cistern.  We  therefore  reject  the  result  6|  as  being  inad- 
missible; the  other  result,  75,  is  admissible,  however,  and  satisfies  the 
requirements  of  the  problem.  Hence  it  takes  the  smaller  pipe  75 
minutes,  and  therefore  the  larger  one  60  minutes  to  fill  the  cistern 
alone. 

2.  A  merchant  selling  some  damaged  goods  for  $72, 
finds  that  his  loss  per  cent,  is  \  of  the  number  of  dollars 
the  goods  cost.     Find  the  cost  of  the  goods. 

Let  ;*:= number  of  dollars  the  goods  cost;  then  5= the  loss  per  cent. 
Therefore,  ^^=the  entire  loss. 

By  the  statement  in  the  problem,  we  have 

5c2+21600=300x, 
^8-300x=-2160a 
;*r2-300x+22500=r900, 
3C-150=±30, 
5C=:180orl20. 
Each  of  these  answers  fulfills  all  the  requirements  of  the  problem, 
and  each  is  admissible. 

3.  One  of  two  numbers  is  f  of  the  other  one  and  the 
sum  of  their  squares  is  208.     Find  the  two  numbers. 

4.  Divide  the  number  60  into  two  such  parts  that  the 
quotient  of  the  greater  divided  by  the  less  may  equal  one 
more  than  twice  the  less. 

5.  A  merchant  bought  a  quantity  of  cloth  for  $120;  if 
he  had  bought  6  yards  more  for  the  same  sum,  the  price 
per  yard  would  have  been  $1  less.  How  many  yards  did 
he  buy,  and  what  was  the  price  per  yard  ? 

6.  A  merchant  sold  two  pieces  of  cloth  which  together 
contained  40  yards,  and  received  for  each  piece  twice  as 
many  cents  per  yard  as  there  were  yards  in  the  piece. 
For  the  smaller  piece  he  received  ^  as  much  as  for  the 
larger  one.     How  many  yards  were  there  in  each  piece  ? 


I  go  UNIVERSITY   ALGEBRA. 

7.  A  merchant  sold  some  goods  for  $39,  and  in  so 
doing  gained  as  much  per  cent,  as  the  goods  cost  him. 
What  was  the  cost  of  the  goods  ? 

.8.  Find  a  number  such  that  3  more  than  twice  the 
number  multipHed  by  3  less  than  twice  the  number  may- 
give  a  product  of  112. 

9.  A  man  traveled  105  miles,  and  then  found  if  he 
had  gone  2  miles  less  per  hour  he  would  have  been  6 
hours  longer  on  his  journey.  How  many  miles  did  he 
travel  per  hour? 

10.  A  man  bought  two  farms  for  $2800  each;  the 
larger  contained  10  acres  more  than  the  smaller,  but  he 
paid  $5  more  per  acre  for  the  smaller  than  for  the  larger. 
How  many  acres  were  there  in  each  farm  ? 

11.  The  length  of  a  rectangle  is  10  feet  more  than  the 
breadth,  and  the  area  is  600  square  feet.  Find  the  length 
and  breadth  of  the  rectangle. 

12.  A  flower  bed  9  feet  long  and  6  feet  wide  has  a 
path  around  it  whose  area  is  equal  to  the  area  of  the  bed 
itself.     What  is  the  width  of  the  path? 

13.  A  number  consists  of  two  digits.  The  digit  in 
unit's  place  being  the  square  of  the  digit  in  ten's  place, 
and  if  54  be  added  to  the  number  the  digits  are  reversed 
in  order.     What  is  the  number? 

14.  Find  a  number  such  that  if  it  be  added  to  94  and 
again  subtracted  from  94  the  product  of  the  sum  and  dif- 
ference thus  obtained  shall  be  8512. 

15.  Find  a  number  such  that  if  its  third  part  be  multi- 
plied by  its  fourth  part  and  to  the  product  5  times  the 
number  be  added  the  sum  exceeds  200  b}^  as  much  as  the 
number  required  is  less  than  280. 


QUADRATIC    EQUATIONS.  I91 

16.  A  man  bought  a  horse  and  sold  it  again  for  $119, 
by  which  means  he  gained  as  much  per  cent,  as  the  horse 
cost  him  dollars.  How  many  dollars  did  the  horse  cost 
him? 

17.  The  combined  area  of  two  squares  is  962  square 
feet,  and  a  side  of  one  square  is  18  feet  longer  than  a  side 
of  the  other.     What  is  the  size  of  each  square  ? 

18.  A  square  field  contains  a  number  of  square  rods 
equal  to  260  more  than  32  times  its  perimeter.  How 
many  rods  in  one  side  of  the  square? 

19.  The  sum  of  the  ages  of  a  father  and  son  is  85 
years,  and  ^  of  the  product  of  their  ages  in  years  exceeds 
five  times  the  father's  age  by  200  years.  What  is  the 
age  of  each  ? 

20.  What  is  the  price  of  oranges  when  10  more  for 
$1.20  lowers  the  price  one  cent  each  ? 

21.  A  certain  number  is  the  product  of  three  consecu- 
tive whole  numbers,  and  if  it  is  divided  by  each  one  oi 
these  three  factors  in  turn  the  sum  of  the  three  quotients 
thus  obtained  is  767.     What  is  the  number? 

22.  The  sum  of  the  squares  of  three  consecutive  odd 
numbers  is  83.     What  are  the  numbers? 

23.  The  sum  of  the  squares  of  four  consecutive  even 
numbers  is  120.     What  are  the  numbers? 

24.  Divide  the  number  18  into  two  such  parts  that 
their  product  shall  exceed  30  times  their  difierence  by  20 

25.  In  a  bag  which  contains  60  coins  of  silver  and  gold 
each  silver  coin  is  worth  as  many  cents  as  there  are  gold 
coins,  and  each  gold  coin  is  worth  as  many  dollars  as 
there  are  silver  coins,  and  the  whole  is  worth  1505.  How 
many  gold  and  how  many  silver  coins  in  the  bag? 


192  UNIVERSITY   ALGEBRA. 

26.  A  man  bought  a  number  of  horses  for  $10000 ; 
each  cost  four  times  as  many  dollars  as  there  were 
horses.     How  many  horses  did  he  buy? 

27.  A  room  whose  length  exceeds  its  breadth  by  8  feet 
is  covered  with  matting  4  feet  wide,  and  the  number  of 
yards  in  length  of  the  matting  exceeds  f  the  number  of 
feet  in  breadth  of  the  room  by  20.  Find  the  length  and 
breadth  of  the  room. 

28.  There  are  two  numbers  whose  difference  is  7,  and 
half  their  product  plus  30  is  equal  to  the  square  of  the 
smaller  number.     What  are  the  numbers  ? 

29.  A  and  B  start  together  on  a  journey  of  36  miles. 
A  travels  one  mile  per  hour  faster  than  B  and  arrives 
three  hours  before  him.     Find  the  rate  of  each. 

30.  Two  workmen,  A  and  B,  are  engaged  to  work  at 
different  wages.  A  works  a  certain  number  of  days  and 
receives  $27,  and  B,  who. works  one  day  less  than  A, 
receives  $34.  If  A  had  worked  two  days  more  and  B 
two  days  less,  they  would  have  received  equal  amounts. 
Find  the  number  of  days  each,  worked. 

EQUATIONS  SOIvVKD   I,IKK   QUADRATICS. 

278.  Some  equations  which  are  not  quadratics  may 
be  solved  by  the  methods  explained  in  this  chapter.  We 
have  had  such  equations  as  ^2-— 13^+36=0,  and  have 
seen  that  such  equations  are  easily  solved.  Now  it  is 
plain  that  we  can  use  some  other  symbol  in  place  of  x  to 
designate  an  unknown  number.  Thus  we  might  have 
an  equation  in  which  y'^  stands  in  place  of  x,  and  of 
course  y^  in  place  of  x'^ ,  and  the  equation  would  be 
y-13j/2+36=0, 


QUADRATIC    EQUATIONS.  1 93 

from  which,  by  solving  in  the  usual  way,  regarding  y'^ 
temporarily  as  the  unknown  number,  we  obtain 

jj/2=4  or  9. 
Hence  j/=±2  or  ±3. 

In  a  similar  manner  we  could  treat  equations  in  which 
more  complex  expressions  stand  in  place  oi  x'^  and  x  in 
the  equations  before  used,  but  whatever  expression  stands 
in  place  of  x  the  square  of  that  expression  must  stand  in 
place  of  x*^,  else  the  equation  cannot  be  solved  by  the 
methods  of  this  chapter. 

KXAMPI,:^S. 

Solve  the  following  equations : 

1.  :r4 -29^2 +  100=0.  3.^4_i7y  +  ie=0. 

2.  jt:^ -35:^3 +216=0.  4.  J/+ 81/^+15=0. 

5.  (^+3)«-28(^+3)3+27=0. 

10.  (;r2-5;i;+6)2-14(;ir2-5;i;+6)-24=0. 

13— U.  A. 


CHAPTER  XIII. 

THEORY  OF   QUADRATIC    EQUATIONS   AND   EXPRESSIONS. 

279.  Equations  and  Expressions.  When  all  the 
terms  of  a  quadratic  equation  are  transposed  to  the  left 
member,  the  right  member  is  zero,  and  the  equation  takes 
the  form  ax^  +  dx+c=0;  or,  if  the  equation  be  divided 
through  by  the  coefficient  of  x^y  it  takes  the  form 
x^  +px+q=0.  The  left  member  of  either  of  these  equa- 
tions is  a  Quadratic  Expression. 

Since  a  root  of  an  equation  has  been  defined  as  any 
expression  which  substituted  for  the  unknown  number 
satisfies  the  equation,  therefore  it  is  evident  from  either 
of  the  forms  ax^  +  dx -j- c=0  or  x'^+px-\-q=^0  that  a  root 
of  a  quadratic  equation  may  also  be  defined  as  any  ex- 
pression which  substituted  for  the  unknown  number  in  a 
quadratic  expression  causes  that  expression  to  vanish;  that 
iSy  to  equal  zero. 

Thus,  the  equation  jr^— 3;ir=10,  whose  roots  are  5  and 
—2,  when  placed  in  the  form  of  a  quadratic  expression 
equal  to  zero,  becomes  ^2--3;r— 10=0.  It  is  now  seen 
that  the  roots  are  such  numbers  that,  when  substituted 
for  X,  cause  the  expression  to  vanish.  For  the  expres- 
sion is  x'^—Zx—\^,  and  putting  5  for  x  it  becomes 
25—15—10,  which  is  zero.  Putting  —2  for  ;i:  the  ex- 
pression becomes  4+6—10,  which  is  also  zero. 

If  any  other  number  than  a  root  is  put  for  x  the  ex- 
pression will  not  vanish  ;  thus  when 

^=—4,  the  expression  becomes  16  +  12  — 10  or      18, 

;tr=— 3,  the  expression  becomes    9+  9 — 10  or       8, 

i»=  — 2,  the  expression  becomes    4+  6—10  or       o, 


THEORY    OF   QUADRATICS.  195 

x=  —  l,  the  expression  becomes    1+  3— 10  or  -—  6, 

x=  0,  the  expression  becomes   0+  0—10  or  —10, 

x=  1,  the  expression  becomes    1—  3—10  or  —12, 

x=  2,  the  expression  becomes   4—  6—10  or  —12, 

x=  3,  the  expression  becomes    9—  9— 10  or  —10, 

x=  4,  the  expression  becomes  16—12—10  or  —  6, 

x==  5,  the  expression  becomes  25  —  15—10  or        o, 

x=s  6,  the  expression  becomes  36—18—10  or       8. 

280.  Factors  of  a  Quadratic  Expression.  We  have 
already  factored  some  quadratic  expressions.    Let  us  now 
take  the  general  quadratic  expression 
x^+px+q. 

Adding  and  subtracting  ^  we  obtain 

or  (^2+^^+^^)_(^_^) 

or  writing  this  as  the  difference  of  two  squares,  we  obtain 

(-l)-(^^^)■ 

Factoring  this  expression,  we  obtain 


From  this  it  is  evident  that  a  quadratic  expression  can 
be  resolved  into  the  product  of  two  linear  factors;  i.  e. ,  two 
factors  each  of  which  is  of  the  first  degree  with  respect  to  the 
unknown  number. 

281.  If  we  should  solve  the  equation  ^^+/jr+^=0 

p        [p^ 
we  would  find  its  roots  to  be  —■^±:^~ — q.     Repre- 
senting these  roots  by  r^  and  rg  respectively,  we  have 


196  UNIVERSITY   ALGEBRA. 


»=2-V4      ^- 


Comparing  these  with  the  factors  of  the  quadratic  ex- 
pression x'^-\-px-\-q  obtained  in  Art.  280  we  see  that  the 
above  factors  are  {pc—r^{x—r^.  Therefore ,  if  the  roots 
of  a  quadratic  equation  are  r-^  and  r^ ,  the  equation  may  be 
written  in  the  form  (x — r{){x — r2)=0. 

It  follows  at  once  from  this  statement  that  if  all  the 
terms  of  a  quadratic  equation  are  transferred  to  one  m^ember 
that  member  is  exactly  divisible  by  x  minus  a  root. 

Since  with  the  meanings  given  to  r^  and  r,^.  ^^  roots  of 
x'^  +px+q=0  are  the  same  as  those  of  (;t:— ri)(^— r2)=0 
it  follows  that  the  form  (x—r-^)(x—r2')=0  may  be  used 
interchangeably  with  x'^  +px+q=0  to  represent  any  quad- 
ratic equation, 

282.  Number  of  Roots.  Representing,  as  before, 
the  roots  of  x^+px-{-q=0  by  r^  and  ^3,  the  values  given 
in  Art.  275  show  that  there  are  two  roots  to  any  quad- 
ratic equation,  but  for  certain  values  of  p  and  q  these 
two  values  are  exactly  the  same.     This  is  the  case  when 

^ — ^=0,  for  then  the  second  term  of  each  root  reduces 

to  zero  and  each  root  of  the  equation  reduces  to  — ^.    In 

this  case  there  is  really  only  one  value  of  x  that  will  sat- 
isfy the  equation.  Instead,  however,  of  saying  that  there 
is  only  one  root  of  the  equation  we  say  that  there  are  two 
roots,  but  that  they  are  equal  to  each  other.  Of  course 
this  is  only  another  way  of  saying  that  there  is  but  one 
value  of  X,  but  further  along  it  will  be  apparent  that  there 
is  a  great  advantage  in  speaking  of  two  equal  roots  rather 
than  one  root. 


THEORY   OF   QUADRATICS.  1 97 

For  certain  values  of  p  and  q  the  number  under  the 
radical  is  negative  and  the  roots  are  imaginary.  (See 
Art.  196.)  In  this  case  there  is  710  rea/^yalne  of .;«;  which 
will  satisfy  the  equation.  Instead,  however,  of  saying 
that  there  is  no  real  root  we  say  that  there  are  two  roots, 
but  they  are  imaginary.  Of  course  this  is  only  another 
way  of  saying  that  there  is  no  real  value  of  .;*:,  but  further 
along  it  will  be  apparent  that  there  is  great  advantage  in 
this  form  of  expression.  With  the  understanding  just 
explained  about  equal  and  imaginary  roots,  we  may 
say  that  every  quadratic  equatio7i  with  one  unknown  num- 
ber has  two  roots  a7id  only  two, 

283.  Sum  of  Roots.  Representing,  as  before,  the 
roots  of  x'^-\-px-\-q=^  by  r^  and  r^  we  have  found 


^--l+V?-^' 


p     \p'' 

By  addition  we  find     r^-}-r2  =  —p. 

Therefore  whe7i  a  quadratic  equation  is  in  the  form 
x^-\-fix+q=0  the  coefficie7it  of  x  with  its  sign  cha7iged  is 
equal  to  the  sum  of  the  roots, 

284.    Product  of  Roots.     As  in  the  previous  article 

P  ,      fp^ 
we  have  ^^"""2      \^ — ^* 


P        IP' 


By  multiplication,  recognizing  the  product  of  a  sum  and 
difference,  we  find 

*  By  way  of  distinction,  a  number  that  is  not  imaginary  is  called  real. 


198  UNIVERSITY    ALGEBRA. 


— (-l)-(Vl^-)=?-(^-)- 

Therefore  when  a  quadratic  equation  is  in  the  form 
x^'j-px-{-q=0  the  term  not  containing  x  is  equal  lo  the 
product  of  the  two  roots. 

In  any  equation  the  term  not  containing  the  unknown 
number  is  called  the  Absolute  Term.  Therefore  when 
a  quadratic  equation  is  in  the  form  x'^-\-px-j-q=0  the  abso- 
lute term,  is  equal  to  the  product  of  the  two  roots, 

285.  Upon  the  results  reached  in  the  two  preceding 
articles  we  may  found  a  new  method  of  solving  the  quad- 
ratic equation  x'^+px-k-q^^. 

We  have  ^1+^2  =  —p\  (1) 

r^r.,^q,  (2) 
Squaring  (1)  we  obtain 

r2+2rir2  +  rl=/2.  (3) 

From  (2),  4^1^2=4^.  (4) 

Subtracting,         rl-l?^  r^-\-r\=p''-4.q.  (5) 

Taking  square  root,  rj  —r^  =  Vp'^  —iq.  (6) 

ri  +  r,  =  -p. (7) 

Adding,  2r^  =  -p+l/p''-4q,  (8) 

Hence,       r,»-|+Jv^^34^=-|+^ZEl£ 

Subtracting  (6)  from  (7),       2r2  =  —p—Vp'^—ig. 
Hence,       r,=  -|-iv//^^=-|-y^-y. 


THEORY    OF    QUADRATICS.  I99 

DISCUSSION  OF  "run  roots. 

286.     Real   and   Imaginary  Roots.    The  roots  of 

x^+px+g=0  are 


^^=-I+Vt^^> 


and  ^2=~|-y^ — g- 

From  the  distinction  between  real  and  imaginary  ex- 
pressions it  follows  that  these  two  roots  are  real  when 
the  expression  under  the  radical  sign  is  positive,  and 
imaginary  when  the  expression  under  the  radical  sign  is 

negative.     Therefore  the  two  roots  are  real  when  ~ — g 

is  positive;  that  is,  when  ^>g;  and  the  two  roots  are 

imaginary  when  ^ — q  is  negative ;  that  is,  when  ^<^. 

If  the  quadratic  equation  were  given  in  a  different  form 
we  could  still  find  when  the  roots  would  be  real  and  when 
imaginary,  for  we  simply  have  to  solve  the  equation  and 
notice  what  expression  appears  under  the  radical  sign, 
and  we  conclude  that  the  roots  will  be  real  when  this 
expression  is  positive,  and  imaginary  when  this  expres- 
sion is  negative. 

BXAMPI.KS. 

Solve  the  following  equations  and  determine  when  the 
roots  are  real  and  when  imaginary: 


I. 

x^-2ax-h2d=0. 

6. 

ax'^—4:bx—4:=^0. 

2. 

ax^-2bx+Zc^0. 

7. 

x^-iax+b^O. 

3. 

a       a 

8. 

x^—4x-{-a=^0. 

4. 

2x'^+Sax=^55. 

9 

a'^x^-}-4bx—4:C=0. 

5. 

ax'^+Ux+^ab^O. 

10. 

2ax'^  +  Sbx—4:abc==0, 

200  UNIVERSITY    ALGEBRA. 

287.    Equal  Roots.     In  order  that  the  above  values 
of  r^  and  rg  may  be  equal  to  each  other  we  must  have 

If  this  value  of  g  be  substituted  for  g  in  the  original 
equation,    that    equation    becomes    x^+px+^=0,    or 
^i=n      Therefore  when  a  quadratic  equation  has 


('+l)=» 


equal  roots  and  all  its  terms  are  in  the  left  member,  that 
member  is  a  perfect  square. 

In  the  preceding  article  we  found  that  the  roots  of 
x^  -\-px+g=0  are  real  when  ^ — q  is  positive,  and  imag- 
inary  when  -t — g  is  negative,  and  in  this  article  we 
have  found  that  the  roots  are  equal  to  each  other  when 
I — ^  ^^  ^^^^*  ^^  ^^^^  appears  that  the  case  of  equal 
roots  is  just  on  the  dividing  line  between  real  and  imag- 
inary roots. 

Kxampi,e;s. 

Determine  when  the  roots  of  each  of  the  following 
equations  are  equal  to  each  other. 

1.  x^—Aax—5d=0.  4.  x'^—2x+a=0. 

2.  x^-ax+2d=0.  5.  x^+--l=0, 

a     b 

3.  x'^-\-2ax-\-ab^=Q,  6.  jt^— -^— ^=0. 


THEORY    OF    QUADRATICS.    ,  20I 

288.    Roots   Numerically  Equal  but  of  Opposite 

Signs.     In  order  that  the  above  values  of  r-^  and  ^3  may 
be  numerically  equal  but  of  opposite  signs  we  must  have 


-IW^--IW1^- 


•  •  2     2 

289.  Roots  Real  and  Numerically  Equal  but  of 
Opposite  Signs.  In  the  preceding  article  we  found 
that  the  roots  are  numerically  equal  but  of  opposite  signs 
when  p=0.  If,  in  addition,  the  roots  are  to  be  real,  the 
expression  under  the  radical  sign  must  be  positive ;  but 
when  p=0  the  expression  under  the  radical  sign  reduces 
to  —q,  and  for  —q  to  be  positive  it  is  necessary  for  q  to 
be  negative.  Hence  ifi  order  that  the  roots  of  x'^  -\-px-j-q=0 
may  be  real  and  ntimerically  equal  but  of  opposite  signs  it 
is  necessary  that  p=0  and  that  q  be  negative, 

KXAMPLKS 

Find  when  the  roots  of  the  following  equations  are 
real  and  numerically  equal  but  of  opposite  signs. 

1.  2x'^+4:ax—U=0,       4.  ^x'^-^ax=a'^—b'^, 

2.  ax'^—2abx+bc=0,      5.  x'^—2{a—b)x-\-b'^—2ab=0 

3.  x''-+^ax—4:=0.  6.  bcx'^  +  (^b'^-^c'^)x-'bc=^0, 

290.  One  Positive  and  One  Negative  Root.  When 
an  expression  consists  of  two  terms,  that  term  which  has 
the  greater  numerical  value,  that  is,  the  value  differing 
most  from  zero,  is  called  the  Dominant  Term.  Now  it 
is  plain  that  when  the  dominant  term  is  positive  the 
whole  expression  is  positive,  and  when  the  dominant 
term  is  negative    the    whole    expression    is   negative. 


202  UNIVERSITY   ALGEBRA. 

Applying  this  principle  to  the  case  of  the  values  of  r^ 
and  ^2  found  above  it  is  plain  that  if  we  wish  one  root 
to  be  positive  and  one  negative  the  part  containing  the 
radical  sign  must  be  the  dominant  term. 


4 


^-q>-^^,.:i^-g>i^;.:-g>0. 


.'.  ^  is  negative.     Therefore  the  equation  x^+px+g=0 
has  one  positive  ayid  one  negative  root  when  q  is  negative, 

291.  Both  Roots  Negative.   In  order  that  the  values 
of  r^  and  ^3  found  above  may  both  be  negative  it  is  plain 

that  —~  must  be  the  dominant  term  and  must  be  nega- 

P 
tive.     Evidently,  for  — ^  to  be  negative  /  must  be  pos- 

itive,  and  for  — ^  to  be  the  dominant  term  we  must  have 
— ^  numerically>'%/=^^^ — q, 

...  ^>^-q;  ..0>-g; 
,'.  —q  is  negative;  /.  q  is  positive.     But  q  must  be  less 
than  ^  in  order  that  the  roots  may  be  real.     Therefore 
hath  roots  of  the  equation  x^-{-px+q=0  are  negative  when 
both  p  and  q  are  positive  and  ^^q. 

292.  Both  Roots  Positive.   In  order  that  the  values 

of  r^  and  f  2  found  above  may  both  be  positive  it  is  plain 

P 
that  — ^  must  be  the  dominant  term  and  must  be  posi- 

P 
tive.    Evidently,  for  —^  to  be  positive  p  must  be  neg- 

tive,  and  for  — ~  to  be  the  dominant  term  we  must  have 


THEORY    OF    QUADRATICS.  203 

.*.  —q  is  negative;  .'.  ^  is  positive.  But  q  must  be  less 
than  ~  in  order  that  the  roots  may  be  real.  Therefore 
both  roots  of  the  equation  x^+px+q=0  are  positive  when 
p  ts  negative y  q  is  positive ^  and  ^>^. 

BXAMPLKS. 

Solve  the  following  six  equations  and  determine  when 
one  root  will  be  positive  and  one  negative. 

1.  ^2  +  6^^_4^=0.  4.  2:r2+6^^— 5^=0. 

2.  ^2+1^+23=0.  5.  x'^—x-U^^, 

3.  ;i;2— |^;r— 5<^=0.  6.   (;r+^)2— ^2^0. 

Find  when  both  roots  of  the  following  four  equations 
are  negative. 

7.  ;*:2  4-10^jr— 10<^=0.  9.  x'^—hax-^a'^ +b=^, 

8.  ;»;2_3^^_2^==0.  10.   (;r— «)2  + (^+^)+^2^0 

Find  when  both  roots  of  the  following  four  equations 
are  positive. 

11.  (^— a)2  +  3^— *=0.        13.  x'^-\-%{x--d)^ah-^^. 

12.  x'^-\-a{x—U)=Q,  14.   (;r+3a)2  +  (;»:+3)+^=0 

15.  Show  that  the  roots  of  the  equation  x'^-^-a'^x-^-b'^^Q 
are  both  negative  when  a'^'>U. 

16.  Show  that  the  roots  of  the  equation  x'^—a'^x-\-b'^=^^ 
are  both  positive  when  ^2^2^. 

17.  Show  that  the  equation  x'^—ax—b'^^==^^  has  one  pos- 
itive and  one  negative  root. 

18.  Show  that  the  equation  x*^  +ax—b^=0  cannot  have 
imaginary  roots. 


204  UNIVERSITY   ALGEBRA. 

19.  If  ri  and  rg  are  the  roots  of  x^-i-px+g=0  find  the 
value  of  rl  +r|  in  terms  of  p  and  g, 

^1  +  ^2  =  -/.  (1) 

^i^s==^-  (3) 

From  (1),  r}  +  2r^r2  +  r|=/2. 

and  from  (2),  ^^1^2='^^ • 

Therefore,  rl  +  rl=_p^ -2c^.    > 

20.  Find  the  value  of 1 —  in  terms  of  p  and  q. 

21.  Find  the  value  of  --\ — '^-  in  terms  of  p  and  c 

22.  Prove  that  (r^  —  ^2  )  ^  =/  ^  —  4^ . 

23.  Form  an  equation  whose  roots  are  the  squares  of  the 
roots  of  the  equation  x'^ -\-px-{-q=^. 

24.  Find  the  value  of  a  such  that  the  roots  of  the  equa- 
tion x^—'4:X-^a=0  will  differ  by  2. 

Historical  Note.  The  origin  of  the  solution  of  quadratic 
equations  has  not  been  definitely  traced  to  any  one  man  or  any  one 
race.  It  is  a  curious  fact  that  the  geometric  solution  of  quadratic,  as 
also  of  cubic,  equations  was  invented  before  the  analytic  method. 
The  former  was  known  to  the  Greeks.  (See  Euclid,  book  VI.,  props. 
27-29;  data,  84,  85.  Negative  values  of  roots  were  ignored  by  them. 
Constructions  of  the  positive  roots  of  equations  were  made  by  the 
intersection  with  one  another  of  straight  lines,  conic  sections,  or 
higher  curves.  Among  the  Arabs  and  the  people  of  the  Occident 
down  to  the  time  of  Cardan,  the  geometric  methods  of  the  Greeks 
preponderated. 

The  algebraic  solution  of  quadratic  equations  was  known  to  Dio- 
phantus,  an  Alexandrian  Greek  of  the  fourth  century,  A.  D.,  who 
wrote  a  treatise  on  arithmetic.  He  rejected  —  roots  and  failed  to 
recognize  two  roots  in  quadratic  equations  even  when  both  were  -f-. 
To  the  Greeks  the  idea  of  multiple-valued  solutions  was  entirely 
foreign.  The  first  to  observe  that  a  quadratic  has  two  roots  and  to 
recognize  the  existence  of  absolutely  negative  quantities  were  the 
Hindoos.  The  earliest  complete  method  of  solving  these  equations, 
together  with  their  application  to  practical  problems,  is  found  in  a 


THEORY   OF   QUADRATICS.  205 

Brahmagupta,  work  of  a  Hindoo  astronomer  of  the  seventh  century 
A.  D.  There  is  considerable  resemblance  between  the  writings  of 
Diophantus  and  of  Hindoo  mathematicians.  We  have  reason  to 
believe  that  during  the  time  of  commercial  intercourse  between  Rome 
and  India,  by  way  of  Alexandria,  Diophantus  got  the  first  glimpses  of 
algebraical  knowledge  from  India,  while  the  Hindoos  afterwards  drew 
upon  the  writings  of  the  gifted  Alexandrian.  The  Hindoos  were  no 
mean  mathematicians.  While  the  Greeks  excelled  in  geometry  they 
surpassed  in  arithmetic  and  algebra.  To  the  Brahmins  we  owe  that 
great  invention  the  "Arabic  Notation."  The  Hindoos  solved  prob- 
lems in  indeterminate  analysis  which  taxed  to  the  utmost  the  powers 
of  Euler  and  Lagrange  in  rediscovering  methods  of  solving  them. 


CHAPTER  XIV. 

THEORY  OF  INDICES. 

293.  The  exponents  whicli  we  have  considered  here- 
tofore are  defined  by  the  following  equation : 

a**=aaaa  ,  ,  .  to  n  factors. 
In  other  words,  a"  is  an  abbreviated  way  of  writing  the 
product  of  n  factors  each  equal  to  a, 

294.  It  has  been  proved  that  positive  integral  ex- 
ponents follow  the  five  laws  expressed  by  the  following 

formulas : 

a''a'•=a"+^  Art.  73.  A 

ar-^ar^ar-\  ifn>r,  Art.  97.  B 

{ary^ar^  Art.  122.  C 

(abcy=a''b''(f\  Art.  123.  D 

Art.  123.  E 


\b)  ^r 


We  shall  find  it  convenient  to  refer  to  these  formulas 
^s  A,  B,  C,  D,  and  E,  respectively,  and  we  shall  speak 
of  them  collectively  as  the  Laws  of  Exponents  or 
Indices. 

295.  It  is  plain  that  in  order  that  oT  may  have  a 
meaning  by  the  definition  already  given  the  exponent  n 
must  be  a  positive  whole  number,  but  in  the  general 
investigations  of  Algebra  it  is  very  desirable  that  the 
formulas  used  shall  not  be  thus  restricted.  We  shall 
therefore  endeavor  to  extend  the  meaning  of  a"  so  that 
fractional  and  negative  exponents  may  be  used. 


THEORY    OF   INDICES.  207 

FR ACTION AI,   EXPONENTS. 

1  H 

296.  Since  expressions  like  a''  and  a^  have  no  meaning 
by  the  definition  previously  given  of  a  number  affected 
with  an  exponent,  we  are  at  liberty  to  define  such  expres- 
sions in  the  manner  that  is  most  convenient.  Now  if  we 
can  discover  a  definition  that  will  make  law  A  hold  for 
fractional  as  well  as  integral  exponents,  that  is  the  defi- 
nition we  shall  adopt. 

If  fractional  exponents  can  be  defined  in  accordance 
with  law  A,  then,  r  being  any  positive  whole  number, 
we  ought  to  have 

1     i     1^  ^  r      ^  M-i+^-...  tor  terns 

a'-a'^a''.  .  .  to  r  factors=«'^''  ~  =^a\ 

1 
that  is,  w  ought  to  mean  one  of  the  r  equal  factors  which 

multiplied  together  produce  a ;  that  is,  dr  ought  to  mean 

the  rth  root  of  a.     Further,  n  and  r  being  any  positive 

whole  numbers,  we  ought  to  have 

a^ara^.  .  .  to  r  factors =a^^^***    ''  ""ss^"; 

n 

that  is,  ar  ought  to  mean  one  of  the  r  equal  factors  which 

n 

multiplied  together  produce  a'*;  that  is,  a^  ought  to  mean 
the  rth  root  of  a". 
Thus  we  see  that  71  and  r  being  positive  whole  num- 

n 

bers  it  is  possible  to  give  d^  a  definition  that  will  make 
law  A  hold  for  firactional  as  well  as  integral  exponents. 
Therefore  we  adopt  the  following  definition :  Any  posi- 
tive fractional  exponent  indicates  a  root  of  a  power  of  a 
number;  the  numerator  indicates  the  power  and  the  denom- 
inator the  root. 

This  definition  is  purposely  chosen  to  make  law  A  hold 
for  fractional  as  well  as  for  integral  exponents,  and  we 
shall  presently  see  that  by  this  same  definition  the  other 
laws  of  indices  will  also  hold. 


208  UNIVERSITY   ALGEBRA. 

297.  Root  of  Power  Equal  to   Power  of  Root. 

I,et  J=x.  (1) 

Raising  both  members  to  rth  power,  we  get 

a=x^,  (2) 

because  the  rth  root  of  the  rth  power  of  a  equals  a. 

Raising  both  members  of  (2)  to  the  nth  power,  we  get 

a-=x*'''=(x''y.  (3) 

Extracting  the  rth  root  of  each  member,  we  get 

(a")'=^^  (4) 

because  the  rth  root  of  the  rth  power  of  x""  equals  x". 

Raising  both  members  of  (1)  to  the  nth  power,  we  get 

(^ary=.x^.  (5) 

Hence,  from  (4)  and  (5),  (a**y  =  (a^y;  that  is,  the  rth 
root  of  the  nth  power  of  a  number  equals  the  nth  power  of 
the  rth  root  of  that  number. 

298.  Comparing  this  with  Art.  296  we  see  that  we  have 

n 

found  two  meanings  for  a^\  first,  the  rth  root  of  the  n\\\. 
power  of  a ;  second,  the  ^th  power  of  the  rth  root  of  a. 
This  may  be  written  in  the  form  of  an  equation  as  follows: 

a-=l/a^=(^^)«,  [1] 

or,  writing  exactly  the  same  equation,  but  using  frac- 
tional exponents  instead  of  radical  signs, 

n  11 

a^=  (««)'-= (a'')^  [2] 

Kxampl:^. 

Write  each  of  the  following  sixteen  expressions,  using 
fractional  exponents  in  place  of  radical  signs  : 

I.  l/a.  5.  V~a^.  g.  Vx^,  13.  i^a—b. 

2.  V^^,       6.  (i/a)«.      10.  {flcy.      14.  {i/~^by, 

3.  ^~a^.  7.  Va^.  II.  1^^.  15.  Va'^-b'', 

4. 1^^.        8,  (T/a)5.      12.  {V^y,        16.  i\a-\-by. 


33-  ^^. 

37.  n^- 

34.  li 

38.  ^i 

3. 

35.  ;;^4. 

39-  <^*. 

36.  xT. 

40.  /i^. 

THEORY   OF    INDICES.  209 

Find  the  numerical  value  of  each,  of  the  following  six- 
teen expressions : 

17.  4i  21.  625i  25.  8lt  29.  2561 

18.  27^.  22.  64^.  26.  125i  30.  64^ 

19.  9i  23.  216^  27.  32I  31.  512^ 

20.  lei  24.  16*  28.  81^.  32.  1287-. 

Write  each  of  the  following  expressions  zn  two  ways^ 
using  radical  signs  instead  of  fractional  exponents : 

127  n 

41.  r^.  45.  a^. 

42.  X^,  46.    ^2«. 
1  w+1 

43.  JV'.  47.    -^    ''     • 

-  o  ^+-^ 

44.  <2J»'.  48.  <2  ^  . 

299.  Value  the  Same  whether  Exponent  is  in  its 
Lowest  Terms  or  not. 

Let  71,  r,  and  t  be  any  positive  whole  numbers.  We 
are  to  prove  that  a^=a^i.     We  know  that 

because  the  rt  th  power  of  the  rt  th  root  of  a  number  equals 
the  number  itself.     Taking  the  rth  root  of  each  side, 

Raising  both  sides  to  the  n  th  power, 

Each  side  is  now  a  power  of  a  root.     By  the  meaning  of 
fractional  indices  (Art.  297)  we  write  this 

a'^-a'^f.  [3] 

Therefore,  a  number  with  a  fractional  exponent  has  the 
same  value  whether  the  exponent  is  expressed  in  its  lowest 
ter^ns  or  not. 

14— u.  A. 


2IO  UNIVERSITY   ALGEBRA. 

300.  We  have  defined  fractional  exponents  in  such  a 
way  that  law  A  holds  for  fractional  as  well  as  for  integral 
exponents,  and  we  now  proceed  to  prove  that  with  the 
same  definition  all  the  other  laws  hold  for  fractional  as 
well  as  integral  exponents. 

301.  Fractional  Exponents  follow  Law  B, 

Let  Hy  r,  s,  and  /  be  any  positive  whole  numbers,  so 
that  -  and  -  are  any  positive  fractions.    We  are  to  prove 


-    -  71       S 

d^^d^=ar~,  if  — >- 
r     t 


n  s  nt  sr 

We  know         ar-^a^—an^atr^  by  Art.  299. 

=  {a^^Y^ic^'Y,       by  equation  [2]. 

by  law  B  for  integral  indices,  since  nt  and  sr  are  whole 

71      S 

numbers  and  nf>sr  if  ->-.- 

nt—sr 

■=^a  rt  ^  by  definition,  Art.  296. 

But  this  last  fractional  exponent  is  what  we  would  get 

s  ti 

if  we  should  subtract  -  from  — .     Therefore 

Therefore,  fractioTial  expoTients  follow  law  B. 

302.   Fractional  Exponents  follow  Law  C. 

Let  71,  r,  s,  and  t  be  any  positive  whole  numbers. 

n  ns 

Cask  I.   To  prove  (ary=a^ . 

n  n    n    n 

We  know         Qi^y=a^a^a^ .  .  .  to  ^  factors, 

by  definition  of  an  integral  index. 

^^?+^^.. .to. terms  ^  by  kw  ^. 

ns 

=ar  ^    by  adding  fractional  indices. 
Therefore,  (aO^=a''.  \p\ 


THEORY   OF   INDICES.  2X1 

Cask  II.  To  prove  (a^y=ari, 

ft    1  ni   1 

We  know        (ary=a(riy,  by  Art.  299. 


=  ([«^«0V 


by  meaning  of  a  fractional  index. 

by  law  C  for  integral  indices. 

by  taking  the  /th  root  of  the  /th  power. 

n 

=^ari^      meaning  of  fractional  index. 

«1  n 

Therefore,  [ay^w^'.  [6] 

n    s  ns 

Casb  III.  To  prove  (aO'=^''^ 
We  know        («^)^=  [(^^)^^ 

by  the  meaning  of  a  fractional  index. 

=  [a^y,  by  Case  II. 

=a^^,  by  Case  I. 

« £        ns 

Therefore,  (an'=a''^  [7] 

Therefore,  fractional  exponents  follow  law  C. 

303.   Fractional  Exponents  follow  Law  D, 

n  n    n  n 

To  prove        {abcf—a^b^cr. 

We  know      {abcy=^\{abcYfy 

by  meaning  of  a  fractional  index. 

by  law  D  for  integral  indices. 

by  law  C,  (Art.  302,  Case  I.) 

by  law  D  for  integral  indices. 

n   n  n 

T=za^b^(^,  taking  rth  root  of  rth  power 
Therefore,  {ahcY=arlr&.  [8] 

Therefore,  fractional  exponents  follow  law  D, 


212  UNIVERSITY   ALGEBRA. 

304.   Fractional  Exponents  follow  Lavr  £. 

S)-[©7 

by  the  meaning  of  a  fractional  index. 
Yn  J  t>y  law^  for  integral  indices. 

=  ft-?   law  C,  (Art.  302,  Case  I.) 


To  prove 
We  know 


\br) 


=[©7 


law  E  for  integral  indices* 


=  —     taking  r\h  root  of  rth  power. 
br 

Therefore,  (|f =^  ^  M 

Therefore,  fractional  expone7its  follow  law  E, 

Perform  the*indicated  operations  in  each  of  the  follow- 
ing examples  by  means  of  the  laws  of  exponents,  now 
proved  to  hold  for  negative  and  fractional  exponents. 

I.  d^y.  a'^. 

tfSx  J=^i+f  (by  A)=J'^'^=J^. 

11  ^  A.  JL 

4.  x^xx^,  6.  x^»xa^», 

3  2  ^  L. 

5.  dJx  a^,  n.  a''X  ^2«. 
8.  a'^-^a'^. 

11.  d>aH^-^Aa'^b^,       13.  ^a^-^^a^. 

^1  n  1      n_ 

12.  9^5-^(2^.  14.  ab^-T-a'^b^^ 


2.  x^X-r^. 

3.  ;f5x;ri^. 


2         1 

9.  h^-^m, 

%  8 

ID.   mJ-i-mT^. 


THEORY    OF    INDICES.  21 3 


15.  C^*)"^. 

{J)iz=Jo  (by  C)=a^^  (by  Art.  299). 

i6.   C^^)^. 

18.   (at)F                       20.   [(^r^^^ 

17.  (ki)i. 

19.   (a^^)^,                    21.    (jr6^)^ 

22.    (a^x'^j/^)'^, 
(«3j»;iyl)f =(c3)l(;c2)5(;i)t  (by  Z>)=«5;^f>'i%  (by  C). 


23- 

(aH^^i. 

25.   (36a4j^2^3)i 

27. 

(S2xiyi)i, 

24. 

(ad^)i. 

26.    (^"2-:rt>'^^*. 

\^4/         (^4)6                           <^8 

28. 

(ia^^3^)i 

30. 

il)* 

3.  cy 

34. 

0   <> 

31- 

^^^\i 

Uv 

33-  i-^Y 

35. 

36.   (a^+a^-^l)(ai+a  —  a^). 

We  arrange 

the  work  thus: 

«3+«3_^^ 

—  a^  —  a  —  a^ 

a^  +  2a^  +  a^        -a^ 

37.   (x+2y^-^Sy^)(x—2jfi+Syi). 


38.  (.;r^+;K^)(^^-_y'^. 

39.  (a^-3^"^^-2-+4a-i-<^- 

3  2  12  1 

40.  (a''—2a''+Sa")(2a''—a"'). 


39.   (a^-3^"^^-2-+4a-i-<^-^^<^^)(^^— 2a^^i). 


214  UNIVERSITY   ALGEBRA. 

41.  (2ab^-Zaib^y, 

3  11  3  11 

42.  (x^^xy'^+x^y—y^)'^(jx^---'y^. 

We  arrange  the  work  thus: 

x^^y^  )  x^-xy^+o^y-y^  (  x-\-y 
x^—xy^ • 


x^y-y^ 

xiy  -yi 

It  is  just  as  important  to  keep  dividend  and  divisor  arranged 
according  to  the  powers  of  some  letter  in  case  the  exponents  are 
fractional  as  in  the  case  they  are  integral.  Fractional  and  integral 
exponents  must  take  the  order  of  their  respective  magnitudes. 

44.  (^-l)-i-(^i-l). 

45.  (.ai-'2aJx^+x^)^(ai—2aixi-hx). 

46.  (a^-d^-A+2dici)'^(ai+di-c^. 

2  1 

47.  Find  the  square  root  of  x^+2x'^+l. 

48.  JPactor  x—2x'^y^-{-j/. 

NKGATIVK  :^XPONBNTS. 

305.  If  the  product  of  two  numbers  is  unity,  either  of 
the  numbers  is  called  the  Reciprocal  of  the  other  num- 
ber. Thus  ^  is  the  reciprocal  of  2,  f  is  the  reciprocal  of 
f ,  etc.  In  other  words,  the  reciprocal  of  a  number  is  1 
divided  by  that  number. 

306.  Since  expressions  like  a~"  have  no  meaning  by 
the  definition  previously  given  of  a  number  afiected  with 
an  exponent,  we  are  at  liberty  to  define  such  expressions 
in  the  manner  that  is  most  convenient.  Now  if  we  can 
discover  a  definition  that  will  make  law  A  hold  for  neg- 
ative as  well  as  positive  exponents,  that  is  the  definition 
we  shall  adopt. 


THEORY    OF   INDICES.  21  5 

If  negative  exponents  can  be  defined  in  accordance 
with  law  A  then,  n  being  any  positive  number,  integral 
or  fractional,  and  hence  —n  any  negative  number,  inte- 
gral or  fractional,  we  ought  to  have 

Let  r  be  taken  greater  than  n ;  then  we  know 

r 

Hence  we  ought  to  have    a''a~''=— , 

or  dividing  by  a'',  <2~'*=— • 

1        ^ 
That  is,  ^  "  i^z^p-A/?  to  mean  —  • 

Thus  we  see  that  it  is  possible  to  give  a~'^  a  definition 
that  will  make  law  A  hold  for  negative  as  well  as  positive 
indices.  Therefore  we  adopt  the  following  definition : 
.  Any  nutnber  with  a  negative  exponent  is  equal  to  the 
reciprocal  of  that  number  with  a  numerically  equal  but  pos- 
itive exponent. 

307.  It  follows  necessarily  from  Art.  306  that 


^nh—r^s 


n"h~^r^d~* — = = = = etc 

"^  ^  d'         b'd'     c-'b^d'     a-^'bU-'d'     a-^c-^'  ^^^' 

That  is,  any  factor  7nay  be  transferred  from  one  term  of 

a  fraction  to  the  other  ter^n  provided  the  sign  of  the  exponent 

of  that  factor  be  changed.     Thus. 

Ir^x      b'^x     4"^«-i^2     3i^2^2 


Y ,  etc. 


BXAMPI.KS. 

Find  the  numerical  value  of  each  of  the  following : 

1.  2-1.  4.  10-^  7.  2-^.  10.  1024~t 

2.  4-2.  5.  1-1.  8.  16-i  II.  512-i 

3.  (-2)-^     6.  2-2.  g.  Sri  12.  625-i 


2l6  UNIVERSITY   ALGEBRA. 

1  5  5-2  16-1 

2  ,   l-»  ■      32-5-  7-1 
14.3=,.           i6.g3j.               18.-2^.           20.—,. 

Write  each  of  the  following  expressions  without  using 
negative  exponents : 

21.  x-'^,        25.  ha-^ .  29.  {x+yy^,  33.  2a^;i;-2j^~^. 

22.  x'^y-'^.    26.  3a-2ri     30.  (--;r)-3.      34.  (~a2)-3. 

1_  2^-2  ^4  ^-\jj\ 

24.  —1-^-      28.    _.,  _^'      32.  ^ —       36.  ^  _.,  .  „ — -> 

x^  X  ^y  ^  5^~t<5  ^^  ^  ^~ 

Write  each  of  the  following  expressions  in  one  li7ie:  \ 

^7-  ^*  ^^*  ?^*  "^^^  47=275*       43.  5^.7^- 

„    4  2x-^y^  Ax-^y-^  ab^  \ 

45.  /  ^  4-^-         46.  -3+^+-+--^- 

(^_^)-3^-2  X  X  X       X 

308.  We  have  defined  negative  exponents  in  such  a 
way  that  law  A  holds  for  negative  as  well  as  for  positive 
exponents,  and  we  now  proceed  to  prove  that  with  this 
same  definition  all  the  other  laws  hold  for  negative  as 
well  as  for  positive  exponents. 

309.  Negative  Exponents  follow  Law  B. 

I^et  n  and  r  stand  for  two  positive  numbers,  integral 
or  fractional;  then  —n  and  —r  stand  for  two  negative 
numbers. 


THEORY    OF    INDICES.  21/ 

Case  I.    To  prove  a^'^ar^^ar-'^-^^. 

We  know   a''-^a~^=a''  x  -n;,,  by  properties  of  fractions. 

by  meaning  of  negative  index. 

=  a^-^-''),  bylaws. 

Therefore,  a-^a-=a^-i-r)^  [10] 

Of  course  —{  —  r)  may  be  written  +r,  and  n  —  [  —  r)  may  be  written 

«  +  r.     The  form  n—{—r)  is  kept  merely  to  show  the  subtraction  of 

the  negative  index. 


Cask  II.    To  prove  a~^-^ar^a~ 

We  know    <2~'*-f-a''=a~'*x— ,  by  properties  of  fractions. 

«a~^Xa~'', 

by  meaning  of  negative  index. 

=  «~"~^,  by  law  ^. 

Therefore,  a-^^a^^a-*"-^.  [11] 

Cask  III.    To  prove  a~''-r-a~''=^"'*~(~''). 

_        _        _       1 
We  know  a  ^'■^a  ''=a  ''x-^-^,     properties  of  fractions. 

=^-''x^~(~''), 

by  meaning  of  negative  index. 
=  «-«-(-''),  by  law^. 

Therefore,  a-«-^a--=a-«-(-^).  [12] 

Therefore,  negative  exponents  follow  law  B, 

310.  It  should  be  noticed  in  the  above  demonstration 
of  law  B  that  no  restriction  whatever  is  placed  upon  the 
relative  magnitudes  of  the  exponents  n  and  r.  Conse- 
quently, law  B  is  proved  for  all  kirids  of  exponents^  whether 
n  is  numerically  greater  thafi  r  or  not. 

Thus:  a'^-^a^  =  a-'^)  a'^-^a^'^  =  a-^]  a^-r-ai==a-^) 
a~'^-~a'^  =  a~^\   etc. 


2l8  UNIVERSITY   ALGEBRA. 

311.  Negative  Exponents  follow  Law  C. 

I>t  n  and  r  stand  for  any  two  positive  numbers,  in- 
tegral or  fractional. 

Cask  I.   To  prove  (««)-''= ^-'"•. 

We  know  (^O'^^^T"^'  t>y  meaning  of  negative  index. 

="^?>       t>y  law  C  for  positive  indices. 

=«""'',  by  meaning  of  negative  index. 
Therefore,  {a-"Y=a-*'''.  [13] 

Cash  II.   To  prove  {0-^=0-*^^ 

We  know  (^~'*)''=(-^),  by  meaning  of  negative  index. 

="i^j      by  law  ^  for  positive  indices. 

=a'"'*'',  by  meaning  of  negative  index. 
.  Therefore,  (a~«)''=a-«^  [14} 

Casb  III.    To  prove  (ar'^y^a^'^ 

1 
We  know  (a"")  ''=^  _„^^y  by  meaning  of  negative  index 

=  -z;^»  by  Case  II. 

a 

=«'*'',     by  meaning  01  negative  index. 
Therefore,  («-«)-''=««'•.  [15] 

Therefore,  negative  exponents  follow  law  C, 

312.  Negative  Exponents  follow  Law  D, 

Let  n  be  any  positive  number,  integral  or  fractional. 
To  prove      (adcy^a-^d-'^c'^**. 

We  know  {abcy*"^  .  .  ^^>  by  meaning  of  negative  index. 
g=  ^.^  ^»  by  law Z^  for  positive  indices. 
=—;   t;;  ~'  by  properties  of  fractions. 


THEORY    OF    INDICES.  219 

by  meaning  of  negative  index. 
Therefore,  {abc)-''=a-''h-"cr*'.  [16] 

Therefore,  negative  expone?its   vllow  law  D, 

313.   Negative  Exponents  follow  Law  E. 

Let  n  be  any  positive  number,  integral  or  fractional. 

_  (ay*"  a"" 

To  prove  y  =^.. 

We  know      (7)  —~rT^'  ^7  meaning  of  negative  index. 
\b) 
=~'        t)y  law^  for  positive  indices. 


a"" 

X 

by  reducing  fraction. 

by  Art.  307. 

[17] 

Therefore, 

Therefore,  negative  exponents  follow  law  E, 
EXAMPL:es. 

Perform  the  indicated  operations  in  each  of  the  follow- 
ing examples  by  means  of  the  laws  of  exponents : 

_2.  \_ 

1.  ^8x^-5.  4.   8(2-4  X  3^2.  7.   ^    3X7;^    3. 

2.  ^-i^xr-io.  5.   2/-iXzA.  8.   ^ax-^'K\bx'^ 

3.  ^r'^-^^-^.  6.  x^-^x~^^.  9.  a~^b~*'-^ab~'', 

10.  (-7^-^^-2)(-4a2^-i)(-u:2^2^-i). 

12  3     2  2     3  3         2 

11.  (2a^r^)(a"^3^— 1^3^24- ^T^-¥). 

12.  7^-1^-2^-3-^8^-2^-^^-^.    14.  18^~^^^^-s-^6a'2-^i^-«. 

13.  ^^x^y-'^ z^  -^1  x-^y~^ 2-^^ .    15.  6xiy~^z^-^2x~^j/iz~^, 


220  UNIVERSITY   ALGEBRA. 

i6.   («-3)2.  21.   (c-^)^.                  26.   (x~id^)~^, 

17.  (^-2)-^  22.     (a^^)-*.                        27.     (a-5|^-10)-|. 

18.  (^^)-2.  23.     (^-43V6)-3.            28.    (— 1^3-)-4^ 

19.  (;^^)-3.  24.    (j»;ij/^~^^            29.    (— ^-4)-3. 

20.  (r"4)~^-  25.   (8x^y-^yi.         30.   (— a^)-!. 

31.   (— 8-s^VV^-  32.   (— l^^jj-^)i 


33 
34. 
35 


42.   («2^-i  +  3a3ji;-2)  (4^-1— 5;t:-i+6a.r-2) 


4«x-i—  5fl:2.x— 2+  Qa^x-^ 

43.  (2x~^—^x-^4:x^)  {^x~^—2x~^+Zx~^). 

4  2     2  4  2  2 

44.  {x~^—2x~^y^+y^)  (x~^—y'^). 

45.  (3jr"t~fx"^+4)x2.;i;~i       47.    (.r~^+J^-2)  (^r"^— j/-2) 

46.  (.r~t4.;r~'^+l)(;r"i._l).  48.   (:r^j/+jj/t)  (^i— jz-i). 

49.  (2a^— 3ajtri)(3^~'^+2:r~^(4a^;r"^4-9a~^.;»;'i"). 

_3.  JL  _i  3.  _^i  i 

50.  {X     ^—X~^J''2+X     2j/_j/2)-^(jj;     2— j^2). 

_3  1 


5C   ^y-y^ 
X~iy-y^ 


51.    (:r-'^+2jr-2— 3;t:-i)-T-.(.;i:-2+3.:r-i). 


THEORY    OF    INDICES.  221 

52.  (x-^-y-^)^(x-i-y-^). 

53-   (:r4+^-^  +  4[:r2  4-:r-2]  +  6)-f-(;t:2+:r-2  +  2). 

Arrange  the  terms  of  dividend  thus :  x'^+4x--{-Q-{-4:X-^-\-x-'^. 

54.  (x~i—x-'^—4:X~'i-hQx-^—2x~^)-T'(x~i—4cX~^-\-  2). 

_  1  1_ 

55.  Simplify  ^^ ^  56.  Simplify  ^  ' ^ 

57.  Simplify  [{ahiy^xCa-h-^iy^^. 

58.  Simplify  [«32(^^^3)i(^2^3)i]i. 

59.  Simplify  (2^a-i-S^d)(S^a-2id')-6i(a'^-d^)  +  2iad. 

ZERO   KXPONKNTS. 

314.  Since  an  expression  of  the  form  a^  has  no  mean- 
ing according  to  the  original  definition  of  a  number 
affected  with  an  exponent,  and  since  such  an  expression 
has  not  been  considered  in  the  treatment  of  fractional  or 
negative  exponents,  therefore  we  must  consider  this  form 
of  expression  if  we  wish  our  formulas  to  be  perfectly 
general. 

315.  We  shall  endeavor  to  discover  a  meaning  for  a^ 
which  will  make  law  A  hold  when  one  or  both  of  the 
exponents  are  zero. 

If  zero  exponents  can  be  defined  in  accordance  with 
law  A,  we  ought  to  have 

Therefore  a^  ought  to  mean  unity,  whatever  number  is 
represented  by  a.  Thus  we  see  that  it  is  possible  to  give 
to  <2^  a  definition  that  will  make  law  A  hold  for  zero  as 
well  as  for  positive  or  negative,  integral  or  fractional  ex- 
ponents.   Therefore  we  adopt  the  following  definition : 

Any  expression  of  the  form  a^  (a  being  any  number  not 
zero)  is  equal  to  unity. 


222  UNIVERSITY   ALGEBRA. 

316.  We  have  defined  zero  exponents  in  such  a  way 
that  law  A  holds  for  zero  as  well  as  positive  or  negative,  * 
integral  or  fractional,  exponents,  and  we  now  proceed  to 
prove  that  with  this  same  definition  all  the  other  laws 
hold  for  zero  exponents. 

317.  Zero  Exponents  follow  Law  B, 
Case)  I.  To  prove  a^^a''=a^-''\  i.  e,  or*'. 
Since  a^  =  \,        ,',  a^-^ar=—=^a-^. 

Cask  II.   To  prove  a''~a^=<a^'*~°;  i.  e.  a*", 

a** 
Since  ^^  =  1,         .*.  ^'*-r-^o-_:     _=^»^ 

Cask  III.    To  prove  «'*~a'*= «'*-'*;  i.  e.  a^. 
We  know  that  a^-^r-a*"—!,  and  since  a^  =  l, 

Therefore,  zero  exponents  follow  law  B. 

318.  Zero  Exponents  follow  Law  C 
Cask  I.    To  prove  {a^Y=a^ , 

Since  «o  =  l  and  l''=l,  .'.  {a^y-=a^ , 

Cask  II.    To  prove  («'*)<> =^^. 

Since  any  number  affected  with  a  zero  exponent  equals 
unity,  .*.  {ary=-\  and  aO  =  l;  .-.  {a:y=^a\ 
Therefore,  zero  exponents  follow  law  C, 

319.  Zero  Exponents  follow  Law  D. 

Since  any  number  affected  with  a  zero  exponent  equals 
unity,  .\  {abcy^l.     Also  «o  =  i^  30^1^  ^o=,i^ 

because  each  member  equals  unity. 
Therefore,  zero  exponents  follow  law  D, 


THEORY    OF    INDICES.  223 

320.  Zero  Exponents  follow  Law  E. 

Since  any  number  affected  with  a  zero  exponent  equals 
unity,  .-.  g)=l.     Also  a^^\  and  b^  =  \.  .'.  [-^)=-^> 
because  each  member  equals  unity. 

Therefore,  zero  expo7ients  follow  law  E, 

EXAMPLKS. 

1.  Multiply  ax^-\-a^x^  by  ax^—\. 

2.  Multiply  X,  (^"2-+:r~'2-),  {x^—x'^'),  (^2_^0)^ 

3.  Divide  x''^ ■\- a^ x'^ —^x^  by  ji:"^'— 2. 

4.  Find  the  product  of  {x^''-\-xy-\-y'^*')'^,   (^"^y")'^, 

111 

5.  Multiply  together   (a'^+ad+d'^y,    (a—dy,    {a— by 

ia'^+ab-^-b'^y. 

?        2 

6.  Simplify  -3—^3  X  --4 r 

/y  2 .v-2 


7.  Simplify f 


+  1 


8.  What  must  be  the  relation  between  :r  andjj/in  order 
that  x-\-y^  may  be  the  reciprocal  of  x— j/"^? 

1311  111  JL 

9 .  Divide  a—x+ 4:a^x'^-'4,a^x'^  by  aJ  +  zd^x'^—x 2 . 

10.  Simplify  [(^t4)-^(a-V*)"2-]-2  4. 

^  ...  «+^       ^fa+b\-'^ 

11.  Form  an  equation  whose  roots  are  -—7  and! \ 

12.  If  ^1  and  r2  are  the  two  roots  of  ax'^-\-bx-\-c=0^ 
find  the  value  of  (b-\-aj  ^y^ +  {b+ar^y^  in  terms  of 
a,  b^  and  <:. 


CHAPTER  XV. 

SURDS. 

321.  The  student  has  learned  that  there  are  two  nota- 
tions in  use  for  indicating  the  root  of  an  expression,  one 
notation  using  the  ordinary  radical  signs,  and  the  other 
using  fractional  exponents.  While  it  is  unnecessary  to 
have  two  ways  of  writing  the  same  thing,  yet,  because 
each  notation  has  special  advantages  in  particular  cases, 
the  two  methods  are  retained.  Of  course  the  same  laws 
(namely,  A,  B,  C,  D,  and  £  of  the  last  chapter)  govern 
the  operations  with  roots,  whatever  form  of  notation  be 
used. 

322.  Rational  and  Irrational.  An  expression  is 
Rational  with  respect  to  any  number  or  numbers  when 
the  numbers  named  are  not  involved  in  any  manner  by 
the  extraction  of  a  root.     Thus, 

is  rational  with  respect  to  x,  but  irrational  with  respect 
to  c  and  d;  the  term  Irrational  being  used  in  just  the 
opposite  sense  to  rational. 

323.  A  Surd  is  the  indicated  root  of  a  commensurable 
number  (integral  or  fractional)  if  that  root  cannot  Jpe  ex- 
actly taken;  as  l/f  or  1^'3.  Expressions  like  l/4,  1^8, 
etc. ,  are  said_to  be  in  the  Form  of  a  Surd.  Expressions 
like  V^a,  ^^ ab,  etc.,  are  often  called  surds,  although,  of 
course,  they  are  such  only  when  the  letters  stand  for 
commensurable  numbers  whose  roots  cannot  be  exactly 
taken. 


SURDS.  225 

It  should  be  noticed  here  that  we  make  a  distinction  between  the 
terms  incommensurable  irrational  expression  and  surd,  a  distinction 
which  is  not  always  made.  According  to  the  definition  given  above 
y  2  +  V2,  V  V3,  V  TT,  are  not  surds,  but  they  are  irrational  and  in- 
commensurable. This  limited  meaning  of  the  word  surd  is  convenient 
and  is  growing  in  use. 

324.  Orders  of  Surds.  Surds  may  be  conveniently 
classified  by  their  indices  as  Quadratic,  Cubic,  Quar- 
tic,  Quintic,  .  .  .  n-tic,  etc.,  as  the  case  may  be. 

325.  The  operations  with  surds  depend  upon  prin- 
ciples established  in  the  last  chapter.  For  convenience 
of  reference  we  restate  below  those  principles  which  are 
used  in  the  present  chapter. 

326.  T/ze  rth  root  of  the  product  of  several  numbers  is 
equal  to  the  product  of  the  r  ih  roots  of  the  several  numbers. 

That  is,  i/~^'.  =  i/ai/'hi/~c  [1] 

1  ILL 

because  (abcY^a^b^c'', 

by  equation  [8],  Chapter  XIV. 

327.  The  r  th  root  of  the  quotient  of  two  numbers  is 
equal  to  the  quotient  of  their  rth  roots, 

because  (■i)'^='T'   equation  [9],  Chap.  XIV. 

\bJ        J^r 

328.  The  rtth  root  of  a  number  equals  the  rth  root  of 
the  tth  root  of  the  nujnber. 

That  is,  Va='^~V^  [3] 

i  L  L 

because  a'''=(a^)'-, 

by  equation  [7],  Chapter  XIV. 

16- u.  A. 


226  UNIVERSITY   ALGEBRA. 

329.  The  rtth  root  of  the  ntth  power  of  a  number 
equals  the  rth  root  of  the  nth  power  of  that  number. 

That  is,  i/^=i7^  [4] 

because  a^t=ar^ 

by  equation  [3],  Chapter  XIV. 

330.  The  n  th  power  of  the  r  th  root  of  a  number  equals 
the  r  th  root  of  the  n  th  power  of  that  number. 

That  is,  (i/ar=i/^  [5] 

This  is  equation  [1],  Chapter  XIV. 

REDUCTION   OF  SURDS. 

331.  If  any  factor  of  the  number  under  the  radical  sign 
is  an  exact  power  of  the  indicated  root,  the  root  of  that  fac- 
tor may  be  extracted  a?id  written  as  the  coefficient  of  the 
surd,  while  the  other  factors  are  left  under  the  radical  sign. 


(1)  Thus,  y8=l/4x2^ 

=  T/4y2  by  [1]. 

=  2l/2 


(2)  Also,  1^81  =  1^27  x3_ 

=  1^27^3  by[l]. 

=  3f/3 


(3)  Also,  V\^ax'^  =  i^%x^y.'lax 

=  t^8x~^t2^  by  [1]. 

=  2jt:#'2^ 


r4)  Also,  f'a''+''b=iya"xa''b 

^i^^^i/Vb  by  [1]. 

^aiy^b 


SURDS.  227 

332.  It  is  sometimes  convenient  to  have  a  surd  in  a 
form  without  a  coefficient.  The  coefficient  can  always 
be  introduced  under  the  radical  sign  by  reversing  the 
process  of  Art.  331. 

(1)  Thus,  2v^6=V^22i/6 

=  1/22x6  by[l]. 

=  1/24 

(2)  Also,  50t/50=  1/502 1/50 


=  1/502x50  by  [1]. 

=  y '125000 
(3)  Also,  41/5=1/4^1/5 

=  #'43_x3  by  [1]. 

=  #320 

333.  As  the  same  process  may  evidently  be  applied  in 
any  case,  we  say:  Any  coefficient  of  a  surd  may  be  intro- 
duced  as  a  factor  under  the  radical  sign,  provided  that  the 
coefficient  be  first  raised  to  a  power  equal  to  the  index  of 
the  surd. 

334.  The  expression  under  the  radical  sign  of  any 
surd  can  always  be  made  a  whole  number. 


(1)  Thus,  -^1=1^^1  x|=l/i| 


=r2Vxi8 


=  1/31^1/18  by[l]. 

=il/l8 


(2)  Also,  1^7  =  1^1x1=1/11 


=  l/^xl£ 

=  1^/^  1/14  by  [1]. 


=il/l4 


228  UNIVERSITY   ALGEBRA. 

(3)  Also.  V^Vf4S 


X  ab''-'' 


by[l] 


335.  As  the  same  process  may  evidently  be  applied  in 
any  case,  we  may  say:  The  expression  under  the  radical 
sign  in  any  surd  can  be  made  ifitegral  if  both  numerator 
afid  denominator  be  multiplied  by  such  a  nu7nber  as  will 
render  the  denominator  a  perfect  power  cf  the  indicated  roof.^ 
and  if  then  the  required  root  be  taken  of  the  denominator 
thus  found. 

336.  We  may  change  the  index  of  some  surds  in  the 
following  manner: 

(1)  Thus,  1^4=l^j/4  by  [3]. 

=  V2  since  V^4=2. 


(2)  Also,        VM0=  l/  r  1000  by  [3]. 

=  t/10  since  #"1000=  10. 

(3)  Also,  1^256^2^8  ==-^1/256^2^8  ^y  [3]. 

=  fl6ca^,  since  V236c'^a^  =  16ca^ 

337.  Since  [3]  is  true  in  all  cases,  we  know  that  the 
index  of  a  surd  can  be  lowered  if  the  expression  under  the 
radical  sign  is  a  perfect  power  corresponding  to  some  factor 
of  the  original  radical  index. 


SURDS.  229 

338.  A  surd  is  in  its  Simplest  Form  when  (1)  no 
factor  of  the  expression  under  the  radical  sign  is  a  per- 
fect power  of  the  required  root,  (2)  the  expression  under 
the  radical  sign  is  integral,  (3)  the  index  of  the  surd  is 
the  lowest  possible. 

.  339.  Methods  of  making  the  different  reductions  re- 
quired by  this  definition  have  already  been  explained. 
We  give  a  few  examples. 


(1)  Simplify  ^g-^-^. 

by  Art.  337. 

=l/4ai, 

by  Art.  335. 

(2)  Simplify  1^^. 

^^400  =  |/^ 

by  Art.  337. 

=il/60 

by  Art.  335. 

=|1/15 

by  Art.  331. 

(3)  Simplify  fl^fH- 

mH=|i/f 

by  Art.  337. 

=5l/| 

by  Art.  331. 

=  1/10 

by  Art.  335. 

340.   In  any  piece  of  work  it  is  usually  expected  that 
all  the  surds  will  finally  be  left  in  their  simplest  form. 

BXAMPI.KS. 

Reduce  each  of  the  following  surds  to  its  simplest  form : 


a" 


■■VI-    '■#    =VI    -i_ 


230  UNIVERSITY   ALGEBRA. 

OPERATIONS   ON   SURDS. 

341.  Surds  which  differ  only  in  their  coefficients  are 
said  to  be  Similar.  Thus  6V''2  and  15l/2  are  similar 
surds;  also  fl^f  and  fl^-f-;  also  5^ad^  and  nV ab'^ . 

342.  The  addition  and  subtraction  of  surds  involves 
no  principle  not  already  used  in  the  addition  and  sub- 
traction of  other  expressions,  as  the  following  examples 
show:  _  _  _ 

(1)  Combine  the  terms  of  10l/7~8l/7+5l/7. 

101/7-31/7+51/7=121/7, 
by  the  usual  process  of  addition  of  terms. 

(2)  Combine  the  terms  of  7 1/2 -  1/I8  +  2^/8. 
Putting  each  surd  in  its  simplest  form,  we  have 

7v/2-l/i8  +  2^8=7l/2-3l/2+4l/2 
=8t/2 
by  the  usual  process  of  addition  of  terms. 

(3)  Combine  the  terms  of  5l/4+2l/32-l/l08. 
Putting  each  surd  in  its  simplest  form,  we  have 

6i/4  +  2l/32-l/l08=5l/4+4]/4-3l/4 
=61/4 

(4)  Combine  the  terms  of  fl/f—fl/f. 
Putting  each  surd  in  its  simplest  form,  we  have 

=il/6  _  .    _ 

(5)  Combine  terms  of  11f^ aH—^a'^i/ ^^b  +  ha^a^b. 
Putting  each  surd  in  its  simplest  form,  we  have 

22l/^-8«2i/64^+5al/^ 

=  15«2|K^ 


SURDS.  231 

343.  We  observe  the  advantage  of  reducing  each  of 
the  surds  in  any  given  expression  to  its  simplest  form, 
for  then  it  can  be  told  whether  or  not  some  of  the  surds 
are  similar  to  each  other,  and  consequently  whether  or 
not  they  can  be  combined  ;  for  07ily  similar  surds  can  be 
combi7ied  iyito  a  single  surd. 

344.  The  product  of  any  number  of  surds  of  the  same 
iiidex  can  always  be  expressed  as  a  single  surd  by  means 
of  equation  [1]. 


(1)  Find  the  product  of  t/2  x  1^5  X  Vn , 


l/2xl/5 

ysVn= 

=  l/2x5x7 

=  1/70 

(2) 

Find  the  product 

of  V^x  i/Ts. 

l/2> 

c  v'i8= 

=  V2xl8 
=  6 

:XT^9. 

(3)  Find  the  product 

of  #^54 

by  [1]. 


by  [1]. 


f/54x  1^9=1^54x9  by  [1]. 

=  3l/2 
The  result  should  always  appear  in  its  simplest  form, 

345.  The  quotient  of  two  surds  of  the  same  index  may 
be  expressed  as  a  single  surd  by  means  of  equation  [2]. 

(1)  Find  the  quotient  of  i/28-t-i/7. 

V^28  H- 1/7  =  1/-^  by  [2]. 

=  1/4=2 

(2)  Find  the  quotient  of  1^81-^  l'/6. 

=3l^^J  by  Art.  331. 

=11^4  by  Art.  335. 

The  result  should  always  appear  in  its  simplest  form. 


232  UNIVERSITY  ALGEBRA. 

346.  If  the  product  or  quotient  of  surds  of  different 
indices  is  sought,  the  surds  may  first  be  reduced  to  a 
common  index  by  Art.  337  or  equation  [4]. 

(i)  Find  the  product  of  Vhy,  ^1. 

l/5x  ^4=15/125x1^16 

=1^125x16=15/2000 

(2)  Find  the  quotient  of  1^9^1/3. 

^9^i/3=i5^81h-i5/27 

=r||=l5/3    __ 

(3)  Find  the  product  of  V ab^  x  V a''-b. 

i/^x  -¥'^b=  T^^TH  X  '^'^1^2 


(4)  Find  the  quotient  of  l5/|x2'^'^. 


=iTl6xl6=^ri6 

347.  Any  power  of  a  surd  may  be  expressed  as  a  single 
surd  by  means  of  the  principle  of  Art.  330.    Thus : 

(1)  Square  #"2. 

(^2)^  =  ^25  by  [5]. 

=  ^4 

(2)  Cube  3l/2. 

iZViy=Z\V^^    by  lawZ»  for  indices. 

=27l/2»  by  [5]. 

=54l/2  by  Art.  331. 

The  result  should  always  be  left  in  its  simplest  form. 


SURDS.  233 

348.  Any  root  of  a  surd  may  be  expressed  as  a  single 
surd  by  means  of  the  principle  of  Art.  328.    Thus : 

(1)  Find  the  square  root  of  1^4. 


1/^/4=1^4=1^1/4  by  [3]. 

=  ]^'2 
(2)  Find  the  cube  root  of  |t/3. 


^|v'3=2l^il/3 

by  Art.  331. 

=21^1/^ 

by  Art.  333. 

=21^1/^ 

=2v'r^ 

by  [3]. 

=2l/i=|l/3 

by  Art.  335. 

(3)  Find  the  ^th  root  of  aVb. 

^ai/b=^Va'-b 

by  Art.  333. 

=  Varb 

by  [3]. 

349.  This  last  process  is  a  general  one,  but  if  for  any 
particular  values  of  a,  b,  r,  and  t  this  result  should  not 
happen  to  be  in  its  simplest  form,  it  should  be  so  reduced. 

RATIONAI.IZATION  OF  EXPRESSIONS   CONTAINING  SURDS. 

350.  To  Rationalize  an  expression  is  to  perform  an 
operation  upon  it  that  will  free  the  expression  of  surds. 
Thus  the  binomial  quadradc  surd  3  +  l/2  is  rationalized 
when  multiplied  by  3— 1/2,  for  the  product  is  9—2  or  7, 
which  is  rational. 

351.  Any  multiplier  which  when  applied  to  an  irra- 
tional expression  will  free  the  expression  of  surds  is 
called  a  Rationalizing  Factor.  Thus  3  — 1/2  is  a 
rationalizing  factor  for  the  binomial  surd  8+1/2. 


234  UNIVERSITY   ALGEBRA. 

352.  It  is  often  convenient  to  perform  an  operation 
upon  both  terms  of  a  fraction,  so  as  to  render  either  the 
numerator  or  the  denominator  rational.  It  is  sufficient 
for  present  purposes  to  show  how  this  may  be  done  when 
all  the  surds  are  of  the  second  order.  The  following  are 
examples. 

2 
(1")  Rationalize  the  denominator  of  7=. 

^  ^  2  +  1/2 

Multiplying  numerator  and  denominator  by  2— 1/2, 
we  get  _  _ 

2      _        2(2-l/2)        _  4-21/2 
2  +  t/2"(2  +  V'2)(2-i/2)~4~  (1/2)2 

3 
(2)  Rationalize  the  denominator  of  —7= 7=. 

^  ^  1/5-1/2 

Multiplying  numerator  and  denominator  by  l/5-fV^2, 
we  get  ,  _        _ 

3        _  3(1/5  +  1/2) 

l/5_  1/2  ""  (1/5  - 1/2")  (1/5  + 1/2  ) 

=5(4±i^=^5+l/2. 

353.  This  work  is  based  on  the  very  evident  principle 
that  any  binomial  quadratic  surd  is  made  rational  by  mul- 
tiplying the  surd  by  itself  with  the  sign  of  one  of  its  terms 
changed,  for  the  product  is  then  the  difference  of  two  squares. 

354.  Considerable  labor  is  often  saved  in  computing 
the  value  of  an  irreducible  fraction  if  we  first  rationalize 
the  denominator.     Thus,  to  compute  the  value  of 

3 

1/5- V^2 


SURDS.  235 

to  five  decimal  places,  the  two  square  roots  must  be 
taken  to  at  least  five  decimal  places,  and  the  quotient 
of  0  divided  by  the  differ e7ice  of  these  roots  must  be 
found.  This  division  by  a  five-place  number  will  be 
avoided  if  we  firs^t  rationalize  the  denominator;  for  the 
result  is  Vb-\-v1,  the  value  of  which  is  found  without 
the  ' '  long  division ' '  of  the  former  method. 

355.      Rationalization     of    Trinomial     Quadratic 
Surds.     Any  trinomial_  quadratic  surd  may  be  repre- 
sented by  V a-\-Vb-^V c.    Multiplying  by  V a-\-V b—V c 
we  obtain 
( l/^-f  V~b-\-  Vc^  (Va  +  V~b-  Vc)  =  (Va-\-  Vb)  ^  -  ( j/^)  2 

=  a-{-b—c-\-2Vab,  (1) 

which  is  rational  as  far  as  c  is  concerned.  Now  multi- 
plying this  last  expression  by 

a  +  b—c—2Vab  (2) 

we  obtain     (a  +  b—c+2\/ab)(a-{-b—c—2Vab) 

=  C^-j-b—cy—4:ab,  (3) 

which  is  rational  with  respect  to  a,  b,  and  c.  The  ration- 
alizing factor  for  the  original  trinomial  quadratic  surd 
is  thus  seen  to  be 

(l/^-f  V'^-V'c)  {a-\-b-c-2Vab)  (4) 

The  second  parenthesis  in  (4)  is  composed  of  the  factors 

(l/^-  V"b+  Vc)  (Va-  Vb-  Vc) 
Hence  the  rationalizing  factor  ot  V a-\-V b-\-Vc  may  be 
written 

(V  ^+  V~b—  Vc)  {Va— Vb+ V~c)  (Va—  Vb—Vc) . 
Observe  that  the  terms  of  each  of  the  component  tri- 
nomial factors  of  this  expression  are  those  of  the  given 
surd,  except  the  signs  are  those  exhibited  in  the  scheme 
+  +  -,      +  -  +,      + . 


236  UNIVERSITY   ALGEBRA. 

Now  it  is  evident  that  there  is  no  other  arrangement 
of  signs,  keeping  the  first  sign  unchanged,  than  the 
arrangements  written  in  this  scheme,  except  the  arrange- 
ment +  +  + ,  which  is  the  arrangement  of  signs  in  the 
given  trinomial.     Therefore; 

The  rationalizing  factor  for  any  trinomial  quadratic 
surd  is  the  product  of  all  the  different  trinomials  which 
can  be  made  from  the  original  by  keeping  the  first  term 
unchanged  and  giving  the  sig7is  +  and  —  to  all  the  re- 
maining terms  in  every  possible  order  except  the  order 
occurring  in  the  given  trinomial. 

As  an  example,  find  the  rationalizing  factor  for 

The  above  method  shows  it  to  be 

(V^5-t/7-t/3  )(l/54- 1/7  + 1/3  )(l/5  + 1/7-1/3  ), 
and  multiplying  the  original  trinomial  by  this,  the  ra- 
tionalized result  is  found  to  be  —59. 

356.  Rationalizing  Factor  for  any  Quadratic 
Surd.  The  above  problem  is  capable  of  generalization, 
but  its  proof  need  not  be  given  here.  The  generalized 
statement  is  as  follows : 

The  rationalizing  factor  for  ajiy  polynomial  quadratic 
surd  is  the  product  of  all  the  different  polynomials  which 
can  be  made  from  the  original  by  keeping  the  first  terfn 
unchanged  and  giving  the  signs  +  and  —  to  all  the  re- 
maining terms  in  every  possible  order  except  the  order 
occurring  in  the  given  polynomial, 

BXAMPLKS. 

Rationalize  the  denominator  of  each  of  the  following : 
.       1-1/24-1/3 
^*   1  +  V2-V^3* 


SURDS.  237 

30  2  +  1/^-1/2 


*  2-1/3  +  1/5  _  _*  2-1/6+T/2 

3-  Show  that  j/^l-Xlvr'-^'^'^'^''~'^- 

357.  Rationalization  of  any  Binomial  Surd.  By 
reducing  the  fractional  exponents  to  a  common  denomi- 
nator  any  binomial  surd  may  be  put  in  the  form  <2«=h^«. 
We  shall  show  how  to  rationalize  each  of  these  forms. 

s  t 

I.  To  rationalize  the  form  a^ — b^.  For  convenience  let 
a»=x  and  d^=y;  whence  ««— ^«=;r— jj/.  Now  multiplying 
x—y  by  x^'-^  +  x^'-y+x^-y  + . . .  4-y'~^  (1) 
we  obtain,  by  Art.  137, 

(x—y)  (x''-^+x'*-y+x'*-y  +  ,  .  .+jj/"-^)=;t:''— jj/". 

But  ;i;«— j/"=(a^"— (^-0«=^'— ^'» 

which   is   rational.     Therefore  (1)  is  the  rationalizing 

factor  for  a»—d». 

s_  I  s 

II.  To  rationalize  the  form  a^  +  b^.   As  before,  let  ««=;»; 
t  £      i 

and  b^=y]  whence  a'^-\-bn=^x+y.     Multiplying  x+y  by 

x^^-i-.x^-'^y^^n-z^z__^  .  .±.y"-^  (2) 

the  product,  by  Arts.  138,  141,  is 

(x+y)  (^x''-^—x"-y+x''-y  —  .  .  .ijv'*"^) 

=x"—y"  if  n  is  even, 

^x^'+y"  if  n  is  odd. 

But  x**—y''=  (any—  (^«)"= a^—  b'; 

:r''+j/«==  (a"y-{-(b-y=a'+b'. 

Each  of  these  results  are  rational ;  therefore  (2)  is  the 

£         -£ 

rationalizing  factor  for  a»+b^. 

The  following  examples  illustrate  the  processes  ex- 
plained above. 


233  UNIVERSITY   ALGEBRA. 

2.  i 

(1)  Rationalize  d^  —  r'^. 

With   a  common    denominator  for  the   exponents,    this    oecomes  ^ 
^e_;/6-  whence  n—Q,  j-=4,  /=3;  then  x^d^,  y=r^. 

±  3.  ISL  l_e     3  12.    6.  8    9  iL    !_?.  1_5.'  4  3. 

(2)  Rationalize  6  +  31/^5. 

With  a  common   denominator  for  the  fractional   exponents,    this 

4.  1  4 

becomes    6^  +  (3*X5)^;    whence    ??=r4,    ^=4,    /=!;    then    :>c=6^   and 
y=(34X5)4.     Since  {x-\-y){x^ -x^y  +  xy^^ -y^)=x't-\-y*, 

[64  + (3^X5)4]  [63-62x3X5^4-6X3^X5^-33x5^] 
[64]4  +  [(34x5)4J4=:64-34x5=891. 

FUNCTIONS   OF   SURDS. 

358.    Function   of  a   Number.     A   Function   of  a 

nnmber  is  a  name  applied  to  anj^  mathematical  expres- 
sion in  which  the  number  appears.     Thus, 


2ax,     x—y'^x'^,     -^ — ^^      va—x'^, 

are  all  functions  of  x.  In  the  same  manner  we  speak  of 
functions  of  several  numbers.  The  second  expression 
abovej  may  be  called  a  function  of  x  and  y.  Obviously, 
a  funcl:ion  of  a  number  might  be  otherwise  defined  as 
any  expression  which  depends  upon  the  number  for  its 
value. 

359.     Rational    Integral    Function.     A    Rational 

Function  of  a  number  is  one  in  which  the  number  is 
not  involved  in  a  radical  or  affected  with  an  irreducible 
fractional  exponent.  An  Integral  Function  of  a  num- 
ber is  one  in  which  the  number  does  not  appear  in  the 
denominator  of  a  fraction  or  is  not  affected  with  a  nega- 
tive exponent. 


SURDS.  239 

A  function  may  be  both  rational  and  integral,  in  which 
case  it  is  called  a  Rational  Integral  Function.  If  n  is 
a  positive  whole  number  and  a^,  a^,  a^^  .  .  .  ci-n  stand  for 
any  numbers  whatever,  then 

is  a  general  expression  representing  any  rational  integral 
fmictioii  of  X  of  the  n  th  degree. 

Such  expressions  as  function  of  x,  function  of  a,  func- 
tion of  x+h,  etc.,  are  abbreviated  into  F(x),  F(a)^ 
F{x-\-li),  or  /(-^),  f{ct),f(^x+h'),  or  a  similar  expression. 
It  must  be  kept  well  in  mind  that  F,  /,  etc.,  are  not 
coefficients. 

360.    Function  of  a  Quadratic  Surd.     Let  V^a  be 

any  quadratic  _surd  and  /(l/<3^)  any  rational  integral 
function  of  y_a,  say       _  _ 

a^(y~ay  +  a^{V~ay-'+.  .  .+a^_y a  +  a„.  (1) 

Now  all  even  powers  of  V  ^  in  this  _expression  will  be 
rational,  and  any_odd  power,  say  (V ay"^^ ,  will  reduce 
to  the  form  a''\/ a.  Therefore  (1)  will  reduce  to  a  form 
containing  but  two  different  kinds  of  terms,  and  may  be 
put  in  the  form  A  +  BV  a,  (2) 

where  A  and  B  are  rational  with  respect  to  a.  Whence 
we  say,  every  rational  integral  function  of  va  may  be 
expressed  as  the  sum  of  a  rational  term  and  a  rational 
multiple  of  V a.  _ 

It  follows  frotn  the  above  reasoning  that  f(—l/a)  dif- 
fers from  f^y a  )  only  in  the  sign  of  the  irrational  part, 
so  that  if  _  _ 

fiVa^^^A-^BVa,  (3) 

then  /(-T/^)=^~^l/a;  (4) 

whence  /(T/a)/(-l/^)  =  ^2_^2^^ 

which  is  rational  with  respect  to  a. 


240  UNIVERSITY    ALGEBRA. 

361.  Function  of  any  Surd.     I^et  a'p  be  any  surd 
1  1 

and  f{a^')  any  rational  integral  function  of  a^,  say 

«o(^V  +  «i(^V"'  +  .  .  .+««-,«^  +  ««.  (1) 

1  1 

Now  any  power  of  a^  greater  than  /—I,  say  {a^y"^', 

s_ 

will  reduce  to  the  form  a'^ap,  where  s  is  less  than  p. 
Whence  (1)  may  be  put  in  the  form 

aJ-^^aJ-^+  .  .  .  +^«-i«^V^n,  (2) 

where  A^,  A^,  A^,  .  .  .  are  rational  with  respect  to  a. 

In  the  chapter  on  determinants  it  will  be  shown  that  a  rational- 
izing factor  can  be  found  for  (2),  and  thereby  prove  that  a  rational- 
izing factor  can  be  found  for  any  irrational  expression  whatever. 

362.  If  ^^  ^,  A,  B  are  commensurable  and  V a,  V  b 
are  incommensurable^  then  v  a  cannot  equal  A+BVb. 

Suppose  that  l/a=A+BV^b,  if  possible.  Then  we 
must  have,  by  squaring,  _ 

a=A^-\-2AB\/b-\-BH, 
-A'^-BH 


or 


T/^=^ 


2AB 

But  by  hypothesis  the  right  side  of  this  equality  is 
commensurable.  Thus  we  have  an  incommensurable 
number  equal  to  a  commensurable,  which  is  absurd. 
Whence  Va  cannot  equal  A  +  BV^b. 

Since  A+Bj/b  may  represent  any  rational  integral 
function  of  l/b  (Art.  360),  we  may  say:  One  quadratic 
surd  cannot  be  expressed  as  a  rational  integral  furiction 
of  another, 

363.  If  CL,  b,  py  q  are  commensurable  and  vb  and  V~q 

incommensurable y  and  if  a+V b=p+vq,  then  a=p  a?id 
V'j^l/q. 

If  a  does  not  equal  py  suppose  a=p  +  d.  Substitute 
this  value  for  a  in  the  given  equation,  and  we  have 


SURDS.  241 

or  d+Vb=Vq. 

From  the  last  article  this  equality  cannot  be  true. 
Therefore  a  cannot  differ  from  p]  and  if  a=p,  V b  must 
equal  V q. 

squar:^   and  square  root  of  binomial  quadratic 

SURDS. 

364.  Square  of  Binomial  Surd.  Let  V a-\-V~b  rep- 
resent any  binomial  quadratic  surd.    We  then  have 

The  square_  of  any  binomial  quadratic  surd  is  of  the 
form  A  +  VB.     See  also  Art.  360. 

365.  Square  Root  of  the  Form  a+j/^  Since  the 
square  of  Va  +  }/b  takes  the  form  A  +  l^B,  it  follows 
that  the_square  root  of  a+l/)8  can  be  be  expressed  in  the 
form  V x+Vy  for  some  values  of  a  and  /?.  We  proceed 
to  find  for  what  values  of  a  and  P  this  is  true. 

Suppose  ±:^a+Vp=^\/x+\/y,  (1) 
where  the  positive  roots  are  to  be  taken  in  the  right 
member.     Then  squaring, 

a+V'^=x+2Vxy+y  (2) 

By  Art.  363  we  have       x+y=a;  (3) 

21/^=1/^;  (4) 

or  from  the  last,                  4:xy=p.  (5) 

From  (3),                                y^a—x.  (6) 

Substituting  in  (5),  ix'^  —  Aax^—p.       (7) 

Whence,  solving,                    x=^(a±:\/a'^—fi),  (8) 

and  from  (3),                          jv==i(a=f:l/a2— ^).  (9) 

16— U.  A. 


242  UNIVERSITY   ALGEBRA. 

Therefore,  substituting  these  values  in  (1),  we  have 
v'a-fl/iS=d=(l/i-(a+l/^:^^)  +  V'l-(a-T/a2_^))       [6] 
Similarly  we  would  find 

Va-V~P=±{y\(a+}/^J^fi)^y'^Ca^V^^))  [7] 
Thus  we  see  that  the  square  root  of  an  expression  of  the 
form  a+V  /?  can  be  expressed  as  the  sum  of  two  simple 
surds  if  o-'^ — P  is  a  perfect  square,  and  can  be  expressed  as 
the  sum  of  two  complex  surds  (real  or  imaginary)  for  all 
values  of  a  and  (3. 

KXAMPLKS. 

Express  the  square  root  of  each  of  the  following  by 
means  of  two  simple  surds : 

1.  47-121/15.       _  _ 

Here  a=:47  and  T//?=12l/l5;  whence  iS=2209.  Hence 

/a- t/)S=  d=:(3l/3-2l/5  ). 

2.  87-121/42.  4.  57-121/15. 

3.  17-31/21.  5.  75  +  121/21. 


6.  Prove  that  |(l/3  + 1)2-2  (l/2- 1)2  =  1/59-24 1/6 

7.  Express  in  the  simplest  possible  form  the  value  of 
Vx+\+Vx—\  when  4:X=V a-\--^' 


V  a 
MISCELI.ANKOUS   KXKRCISKS. 

1/12 


1.  Find  the  value  of  ,—        ,—       ,— 

(1  +  1/2)  (1/6-1/3) 

2.  Simplify  l/l2+-jV^75  +  6V^S. 

3.  Simplify 


1  +  1/8+ 1/2- l/27-l/l2+l/75-T^^19  +  6l/g. 

4.  Simplify  1^2r+7l^x  1^^^21-71/2. 

5.  Find  the  value  of  x''-^x-\-l  if  ;r=3-l/3. 


SURDS.  243 

6.  Find   the   value   of  x^-\- 2j^^-{- 22^+ 6xy^  when 

7.  Find  the  value,  when  x=vS,  of  the  expression 

2:r— 1        2.;tr+l 

(x-iy    (x+iy 

8.  Find  the  value  of 

(35l/l0+77v^+63v/3  +  28l/l5)  (i/T04-i/2-t/3) 

9.  Prove  that  the  greater  of  the  two  fractions 

1/3  +  1/2  1/3- ^2 

V  2  + 1^2  + 1/3      t/2  - 1/2-1/3 
exceeds  the  less  by  2  — 1/2. 

10.  Multiply   together  V2x-{-  V2{2x—1)  —-^  and 
-1 V2;i: 

-4=:+i/2(2^-l)-l/2^. 
V  2x 

11.  If  />  and  ^  are  whole  numbers,  find  the  factor  which 

*  1       1 

will  rationalize  aP-\-b^. 

12.  Find  the  rationalizing  factor  for  32"+ 2^,  and  show 
that  the  rationalized  result  is  23. 


l/x+a     \/x—a\f      ^^x^—a^ 


13.  Simplify  {~--=--y==\l 
^V  X — a     V  x-\-ay^ 


-\-ay^y  (x-\-aY  —  ax^ 

14.  Form  an  equation  whose  roots  are -^  and ;= 

3  +  1/5         3-1/5 

15.  Extract  the  square  root  of 

Vlc+Jx--  ^ 

\  X  / —        1 

16.  Find  the  value  of ; =:  when  2jr=  V a  +  -— = 

,/-        /I  Va 

V  X 


'^"V^-^ 


244  UNIVERSITY    ALGEBRA. 

17.  If  2:^=x'^+j/'^,  find  the  simplest  form  of 

V{x+y-\-z)  {x-^y—z)  {2-\-x—y)  (y-^z—x). 

18.  Prove  2  +  1/3  is  the  reciprocal  of  2  — l/S;  and  find 
what  must  be  the_relation  between  the  two  terms  x  and 
\/y  so  that  x-{-  Vy  shall  be  the  reciprocal  of  x—Vy, 

ig.  Prove  that  if />=3  and  ^=5, 

e^-'+pq-"^  -Vqp-^  +g^-^^3g^  +  5 


CHAPTER  XVI. 

SINGI.K  EQUATIONS. 

366.  Roots  and  Factors.  Every  equation  contain- 
ing one  unknown  number  can  be  put  in  the  form 

function  of  x=0 
by  transposing  all  the  terms  to  the  left  side  of  the  equa- 
tion.    If  it  is  an  equation  of  the  first  degree  it  will 
always  reduce  to  the  form  (Art.  233) 

ax-\-b=0  or  x—r^=^0,  (1) 

in  which  r^  stands  for  any  number  whatever,  positive  or 
negative,  integral  or  fractional,  commensurable  or  incom- 
mensurable. It  is  evident  that  this  equation  has  the  root 
^1  and  no  other.  An  equation  of  the  first  degree  may  be 
defined  as  an  equation  which  can  be  placed  in  the  form 
of  a  rational  integral  linear  function  of  x  equal  to  zero. 

We  have  seen  (Art.  281)  that  every  quadratic  equa- 
tion can  be  placed  in  the  form 

ax'^  +  bx+c=^0  or  (ix—r^)(^x—r,^)=0,  (2) 

which  has  the  two  roots  r^  and  ^2  and  no  others.  Thus 
every  quadratic  equation  can  be  placed  in  the  form  of 
the  product  of  two  rational  integral  linear  functions  of  x 
equal  to  zero. 

It  will  be  proved  in  a  subsequent  chapter  that  every 
cubic  equation  can  be  put  in  the  form 

ax^-\-bx'^+cx+d=-0  or  {x—r^){x—r^)(x—r^)=^0,  (3) 
and  that  it  has  three  roots,  r^,  r^,  r^,  and  no  others. 
That  is,  every  cubic  equation  can  be  placed  in  the  form 
of  the  product  of  three  rational  integral  functions  of  x 
equal  to  zero. 


246  UNIVERSITY    ALGEBRA. 

Like  truthvS  will  be  shown  in  a  subsequent  chapter  to 
hold  for  equations  of  the  fourth  and  higher  degrees. 

367.  We  are  led  to  inquire  what  operations  may  be 
performed  upon  the  members  of  an  equation  without 
modifying  the  number  of  values  of  the  unknown  num- 
ber. Now,  by  the  principles  of  algebra,  a7i  equation  re- 
mains true  if  we  unite  the  same  number  to  both  sides  by 
addition  or  subtraction ;  or  if  we  multiply  or  divide  both 
members  by  the  same  number;  or  if  like  powers  or  roots 
of  both  members  be  taken.  But  these  operations  may 
affect  the  7iu?nber  of  values  of  the  unknown  numbers.  Thus 
the  roots  of  the  equation 

Z{x-h^=x{x-h)-\'x''-1^  (1) 

are  —  1  and  5 ;  for  either  of  these  when  substituted  for  x 
will  satisfy  the  equation.  But  dividing  the  equation 
through  by  x—h,  the  resulting  equation  is 

Z=x+x-\-b,  (2) 

Now  this  equation  is  not  satisfied  for  x=^h,  the  only  root 
being  —1.  Hence,  although  equation  (2)  must  be  true 
if  (1)  is  true,  yet  the  equations  are  not  equivalent  since 
their  solutions  are  not  identical.  One  root  has  disap- 
peared in  the  transformation.  Just  how  this  change 
occurs  will  be  best  seen  after  we  place  (1)  in  the  form 
{x—r^{x—r^=^^.  Since  the  roots  of  (1)  are  — 1  and  5, 
by  the  principle  of  Art.  281  the  equation  is  equivalent  to 

(jr-5)(:i:+l)=0.  (3) 

Now  if  we  divide  this  through  by  x—h  we  remove  the 
factor  in  the  left  member  which  is  zero  for  x^h,  and 
consequently  the  equation  will  be  no  longer  satisfied  for 
x—h.  If  we  divide  through  by  jr+1  the  equation  will 
be  no  longer  satisfied  for  x=  —1. 


SINGLE    EQUATIONS.'  247 

Also  consider  the  equation 

x''~6x+S=0,  (4) 

which  is  satisfied  for  x=2  or  x=4.  Multiplying  both 
members  by  :r— 3.  we  obtain 

Cx-SX^^-^^+8)=0.  (5) 

But  this  equation  is  satisfied  for  either  x=S  or  ;r=2  or 
x=4:.  Hence,  although  multiplying  both  members  of 
(4)  by  x—S  has  not  altered  the  fact  of  the  equality  of 
the  members  of  the  equation,  yet  a  value  of  x  extraneous 
to  the  original  equation  has  been  introduced. 
Again,  the  equation 

2x-l=;r  +  5  (6) 

is  satisfied  only  by  the  value  x=6.  Now  by 'squaring 
both  sides  of  the  equation  we  obtain 

4:x'^-4x-\-l=x''+10x-\-25,  (7) 

which  is  satisfied  for  either  x=6  or  x=—^.  Here, 
obviously,  an  extraneous  solution  has  been  introduced 
by  the  operation  of  squaring  both  members. 

In  a  like  manner  notice  the  effect  of  taking  a  root  of 
both  members  of  an  equation.     Thus  suppose 

4x''  =  (x-6y.  (8) 

This  is  satisfied  for  either  x=2  or  —6.  Taking  the 
square  root  of  each  member,  we  obtain 

2x=x—6,  (9) 

which  is  satisfied  only  by  x=—6.     We  have  lost  one  of 
the  roots  of  the  equation  by  this  transformation.     Equa- 
tion (8)  is  really  not  equivalent  to  (9),  but  to  the  two  " 
equations  f  2x=  +  (-^—6)  \ 

\2x=-(x-6'))  ^     ] 

We  have  given  examples  enough  to  show  that  certain 
operations  upon  an  equation  may  modify  the  number  of 
values  of  the  unknown  number.  Thus  we  see  that  dur- 
ing a  series  of  transformations  which  an  equation  must 
sometimes  undergo  it  is  possible  that  the  values  of  the  un- 


248  UNIVERSITY   ALGEBRA. 

known  number  that  satisfy  the  original  equation  may  all 
be  lost  and  that  any  number  of  new  values  of  the  un- 
known number  may  be  introduced.  Thus  none  of  the 
final  results  may  satisfy  the  original  equation,  and  con- 
sequently they  may  have  no  relation  at  all  to  the  prob- 
lem in  hand.  It  is  now  proposed  to  formulate  certain 
propositions  which  will  enable  us  to  tell  the  exact  place 
in  the  process  of  any  solution  where  roots  may  be  lost  or 
new  ones  may  enter.  We  shall  then  be  able  to  perform 
operations  on  the  members  of  an  equation  if  we  will  note 
at  the  time  their  effect  on  the  solution  and  finally  make 
allowance  for  it  in  the  result.  This  fact  must  be  em- 
phasized*: the  test  for  any  solution  of  an  equation  is  that  it 
satisfy  the  original  equation.  '*No  matter  how  elaborate 
or  ingenious  the  process  by  which  the  solution  has  been 
obtained,  if  it  do  not  stand  this  test  it  is  no  solution; 
and,  on  the  other  hand,  no  matter  how  simply  obtained, 
provided  it  do  stand  this  test,  it  is  a  solution." —  Crystal, 

368.  Legitimate  and  Questionable  Transforma- 
tions. If  one  equation  is  derived  from  another  by  ah 
operation  which  has  no  effect  one  way  or  another  on  the 
solution,  it  is  spoken  of  as  a  Legitimate  Transforma- 
tion or  derivation ;  if  the  operation  does  have  an  effect 
upon  the  final  result,  it  is  called  a  Questionable  Trans- 
formation or  derivation,  meaning  thereby  that  the  opera- 
tion requires  examination. 

369.  Equivalence  of  Equations.  Two  equations 
Such  that  any  solution  of  the  first  is  a  solution  of  the 
second,  and  also  that  any  solution  of  the  second  is  a 
solution  of  the  first,  are  said  to  be  Equivalent.  Thus 
the  equations  x'^  —  ^x—Z, 

2.;r(;i;-4)4-6=0 


SINGLE   EQUATIONS.  249 

are  equivalent,  since  each  is  satisfied  by  the  values  x=  3 
and  x=  1  and  by  no  other  values  of  x. 

One  equation  is  said  to  be  equivalent  to  several  others 
when  any  solution  of  the  first  is  a  solution  of  one  of  the 
latter  equations,  and  any  solution  of  any  of  the  latter 
equations  is  a  solution  of  the  first.     Thus 

X'^=4:X  —  B 

is  equivalent  to  the  two  equations 


\x-S=^0] 


Note  that  the  solution  to  any  equation  consists  merely  of  the  simple 
linear  equations  (of  the  form  xz=a)  which  together  are  equivalent  to 
the  original  equation.  Thus  x=l,  x=2,  x=S,  which  is  the  solution 
of  x^  —  Qx^-{-llx  —  Q=0,  are  equivalent  to  the  latter  equation.  A  solu- 
tion to  an  equation  is  another  equation  or  a  set  of  equations. 

370.  The  transformation  of  an  equation  by  the  addition 
or  subtraction  from  both  members  of  either  a  known  number 
or  a  function  of  the  unknown  number  is  a  legitimate  deri- 
vation. 

An  equation  containing  one  unknown  number,  as  it 
commonly  appears  with  an  expression  on  each  side  of 
the  sign  =,  may  be  generalized  in  thought  by  the  ex- 
pression 

a  function  of  x^=  another  function  of  x, 

or  using  L  to  represent  the  left-hand  side  of  the  equation, 
whatever  it  may  be,  and  J?  to  represent  the  expression  on 
the  right-hand  side,  we  can  represent  the  equation  very 
conveniently  by 

L=^R.  (1) 

Now  suppose  that  T,  which  may  be  either  a  known  num- 
ber or  a  function  of  the  unknown  number,  be  added  to 
both  members  of  the  equation,  making 

L-\-T=R+T.  (2) 


250  UNIVERSITY   ALGEBRA. 

Now  it  is  plain  that  (2)  cannot  be  satisfied  unless  L=Ry 
and  that  it  is  satisfied  if  L^*R.  Therefore,  by  Art.  368, 
the  derivation  is  legitimate. 

Thus:  x^=ix-S 

by  adding  —  4x  to  each  side  is  equivalent  to 

X^-4:X=4:X~3-4:X, 

and  this  latter,  by  adding  4  to  each  side,  is  equivalent  to 
;»r2-4x  +  4— -3+4. 

371.  Transposition  of  terms  from  one  memhe>  to  the 
other^  changing  the  signs  at  the  same  time^  is  legitimate. 
Thus,  a  L=R,  to  pass  to  L—R=r^O  is  merely  subtracting 
R  from  both  members. 

372.  Multiplying  both  members  of  an  equation  by  the 
same  expression  is  legitimate  if  the  expression  is  a  known 
number,  not  zero,  but  questionable  if  the  expression  is  a 
function  of  the  u?iknown  nufnber. 

Represent  the  equation  by 

L=R.  (1) 

Multiplying  both  members  by  T,  we  obtain 

LT=RT.  (2) 

Now  this  may  be  written, 

iL-R')T==0.  (3) 

If  Z  is  a  known  number  not  zero  this  can  be  satisfied 
only  by  the  supposition  that  L—R\  that  is,  the  equation 
is  equivalent  to  (1). 

If  7"  is  a  function  of  the  unknown  number,  then  (3) 
may  be  satisfied  by  any  value  of  the  unknown  number 
that  will  make  7^=0,  as  well  as  for  the  values  that  make 
L=R.  Whence  (3),  on  this  supposition,  is  not  equiva- 
lent to  (1),  but  to  the  two  equations, 


{ ^=0 } 


bINGLE    EQUATIONS.  251 

As  an  example,  suppose  L  is  x^,  R  is  4^—3,  and  T  is  2.     Then 

X^=4:X-^  (1) 

is  equivalent  to  2x'^=^x—Q,  (2) 

But  if  T  is  ^—4,  we  get 

{x-'i)x^  =  {^x-^){x-4:),  (3) 

which  is  not  equivalent  to  (1),  since  it  is  satisfied  by  ^=1,  ^=3, 
x—A,  and  (1)  is  satisfied  only  by  x^l,  x—^.  Also,  if  T  is  x^  —  4:,  the 
new  equation  will  be  satisfied  by  x^—'Z,  x:=z\,  x=2,  x=3. 

373.  If  any  equatio7i  involves  fractions  with  only  knozvn 
numbers  in  the  denominators^  it  is  legitimate  to  clear  of 
fractions.  The  multiplier  in  this  case  is  a  known  number. 

374.  If  any  equation  involves  fractions,  some  or  all  of 
whose  denominators  are  furictions  of  the  unknown  7iumber, 
and  if  these  fractioyis  are  all  in  their  lowest  terms  and  710 
two  denominators  have  a  common  factor,  it  is  legitimate  to 
integralize  the  equatio7i  by  multiplyi7ig  by  the  product  of 
the  deno7ni7iators. 

To  illustrate  the  reasoning,  take  the  equation 

ARC 
^+|-  +  -J-+Z?=0,  (1) 

^1         ^2        ^3 

in  which  the  fractions  are  supposed  to  be  in  their  lowest 
terms  and  X^,  X^,  X^  represent  different  integral  func- 
tions of  the  unknown  number,  and  in  which  A,  B,  C, 
and  ^  are  either  known  numbers  or  functions  of  the  un- 
known number.  Multiplying  by  the  the  product  of  the 
denominators,  we  obtain 

AX^X^^BX^X^  4-  CX^X^  +DX^X^X^=^.        (2) 

Now,  since  X^,  ^2,  X^,  A,  B,  C,  and  D  have  no  com- 
mon factor,  no  common  factor  has  been  introduced 
into  all  the  terms  of  the  equation  by  multiplying  by 
^1X2^3,  and  consequently  no  additional  solutions  can 
appear. 


252  UNIVERSITY   ALGEBRA. 

As  an  example  under  the  above  theorem  take: 

(1)  Solve  IT^+fef=2.  (1) 
These  fractions  are  in  their  lowest  terms  and  their  denominators 

are  prime  to  each  other.  Multiplying  through  by  the  product  of  the 
denominators,  we  obtain 

(3^-l)(7-5c)+(ll-2^)(4;t:-5)=2(ll-2x)(3:r-l).  (2) 

Now  we  can  see  that  although  (1)  has  been  multiplied  through  both 
by  (11  —  2^)  and  (3:^—1),  yet  neither  of  these  has  been  introduced  as  a 
factor  through  the  equation.  Hence  there  is  no  additional  solution 
introduced.  The  roots  of  (2)  will  in  fact  be  found  to  be  4  or  —10, 
which  values  also  satisfy  (1). 

But  an  extraneous  solution  may  be  introduced  if  the  denominators 
are  not  prime  to  each  other,  or  if  some  of  the  fractions  are  not  in 
their  lowest  terms.     Thus 

(2)  Solve  _-=__+__.  (1) 

These  fractions  have  two  denominators  alike,  and  consequently  not 
prime  to  each  other.  Multiplying  through  by  the  common  denom- 
inator, x^—9,  we  obtain 

Sx{x-\-S)=Q{x-S)-\-9{x+d),  (2) 

or  reducing,  x*-2xz=d,  (3) 

whose  roots  are  3  and  —1.  Now  if  we  put  the  original  equation  (1) 
in  the  form  Sx-9_    6 

x-d  "^  +  3 

that  is,  3=^  (4) 

it  is  seen  that  it  is  satisfied  only  for  x=—l.  Hence  a  solution  was 
introduced  in  clearing  (1)  of  fractions.  It  is  easy  to  see  that  (1)  is 
really  equivalent  to  (4),  and  hence  that  in  clearing  (1)  of  fractions  by 
multiplying  by  ^'^  —  9  we  multiplied  by  ^  —  3  when  it  was  not"  neces- 
sary; this  is  where  the  solution  ^=3  was  introduced. 

(3)  Solve  "^'"^-+4^+7=15.  (1) 
Clearing  of  fractions,  we  get 

a:»-5^+6-f4x;2  + 7;»r-12x-21=15^-45.  (2) 

or  x*—5x=—Q;  whence  x=2  or  3. 

Putting  (1)  in  the  form  ^-2+4;c+7=:15  {?>) 

it  is  seen  that  the  value  ^=3  was  introduced  in  clearing  (1)  of  frac- 
tions, since  ^=3  does  not  satisfy  (3).  Equation  (1)  is  not  a  case  of 
Art.  374,  since  the  fraction  is  not  in  its  lowest  terms. 


SINGLE    EQUATIONS.  ^53 

375.  Every  equation  can   be  integralized  legitimately. 
If  the  several  fractions  in  the  equation  are  not  in  their 

lowest  terms,  they  can  be  so  reduced.  Then  these  frac- 
tions can  all  be  transposed  to  one  side  of  the  equation, 
their  least  common  denominator  found,  and  then  added 
together.  This  will  now  give  but  one  fraction  in  the 
equation,  and  when  this  is  reduced  to  its  lowest  terms 
we  shall  have  an  equation  of  the  form 

which  by  Art.  374  will  take  on  no  additional  solutions 

N 
when  multiplied  through  by  D,  since  ^  is  in  its  lowest 

terms  by  supposition. 

376.  The  raising  of  both  members  of  an  equation  to  the 
same  poiver  is  equivalent  to  multiplying  through  by  a  func- 
tion of  the  unknown  number,  and  hence  is  a  questionable 
derivation. 

Take  the  equation  L=R  (1) 

and  raise  both  members  to  the  n\,\i  power,  obtaining 

L^=R\  (2) 

Now  (1)  is  equivalent  to 

L-R^O,  (3) 

and  (2)  is  equivalent  to 

Z'*--i?"=0.  (4) 

But  (4)  can  be  derived  from  (3)  by  multiplying  both 
members  by 

'L"-^+L''-'R+L*'-^R^  +  ,  .  ,+L''R''-^  +  LR''-''+R'''^ 
whence  (2)  is  equivalent  to  the  two  equations 

L=R, 

Thus,  squaring  jc— 2=3^  — 8  will  introduce  the  solution  of  Z  +  ^=0 
or,  in  this  case,  of  4x— 10=0;  that  is,  x=f . 


2  54  UNIVERSITY   ALGEBRA. 

377.  Drnding  both  me^nbers  of  an  equation  by  the  same 
expression  is  legitimate  if  the  expression  is  a  knoivn  num- 
ber^ but  qjiestionable  if  it  is  a  function  of  the  unknown 
number. 

Suppose  both  members  of  the  equation  to  be  divisible 
by  T  and  write  the  equation 

LT=RT.  -  (1) 

Now  if  7"  is  a  known  number  this  equation,  by  Art.  372, 
is  equivalent  to  L—R,  (2) 

whence  division  by  T  would  be  legitimate.  But  if  7"  is  a 
function  of  the  unknown  number,  then  (1)  is  equivalent 
to  the  two  equations  {  L=^R\ 

\  T=Q.  I 
Division  by  T  would  give  us  but  one  of  these,  and  con- 
sequently solutions  would  be  lost.     Hence  the  division 
by  a  function  ot  the  unknown  number  is  a  questionable 
derivation. 

Solve  the  equation         {x  —  ^)x'^=z[4:X—^)[x—^), 
Dividing  by  ^  —  5,  we  get  .r2=4x  — 3. 

Transposing  and  completing  square, 

whence  x^i  or  3. 

Restoring  the  value  lost  by  division  by  ^—5, 
x=zl  or  3  or  5. 

378.  The  extraction  of  the  same  root  of  both  members  of 
an  equation  is  equivalent  to  dividing  by  a  functio7i  of  the 
unknown  nu?nber^  and  hence  is  a  questionable  derivation. 

For  we  may  pass  from     L^^R"*  (1) 

to  L^R  (2) 

by  dividing  both  members  of  (1)  by 

Lr-'^+Lr-^R'\'Lr-''R^+, .  .+l^R''-^+lr''-^+R''-'' 

Hence,  by  Art.  377,  root  extraction  is  a  questionable 
derivation. 


SINGLE    EQUATIONS.  255 

Solve  (2;r-l)2  =  (jt-  +  3)2.  (1) 

Taking  the  square  root  of  each  side, 

2x-l=x-j-3-  (2) 

whence  x=4. 

The  solution  lost  is  x=  — |,  which  would  have  been  avoided  by  writ- 
ing (2)  as  follows:  2^—1=  ±(^+3). 

EXAMPLES. 

Solve  each  of  the  following  equations : 

Dividing  by  x~5,  we  get 

S{x  -  l){x  -  2)=(x+  2)(^  +  3).  (1) 

Expanding,  3x^—dx-\-G=:x^-\-5x  +  Q.  (2) 

Transposing  and  uniting,  2;i?2_  14^^—0.  (3) 

Dividing  by  2x,  ^-7=0,  (4) 

or  ^=7. 

Restoring  values  lost  by  division,  x=5  or  0  or  7. 

2.  ax(cx—Sd')  =  da(Sd—cx). 

3.  a^x(c^x—a^d)  =  (a^d—c'^x)d'^c^, 

4.  a^—x'^^(a—x')(d—c — x'). 

5. \-ax^-j--\-dx. 

a  0 

6.  x'^-^x=^n'^+n. 

Transposing,  x"^  -n^^i^n  —  x. 
Dividing  by  ;c-«,  x  +  n=  —  l; 

whence  xz=  —  n~l. 

Restoring  solution  lost,  x=?i  or  —?t-  1. 

7.  (a-xy—ia-dy^id-xy. 
Dividing  by  ^  -  x, 

{a-x)^  +  {a—x){a-d)-\-{a-3)^  =  {3—x)^. 
Transposing,  {a—x){a  —  d)-{-  {a—dy^ ={d  —  x)^  —{a-  x)^. 
Dividing  by  a  —b,  a—  x-\-a  —  b=z  —  [b  —  x-\-a—x), 

or  3^=3«  or  x^^a. 

Restoring  value  of  x  lost,  x=ia  cr  b, 

8.  (^-4)3  +  (;r_5)3  =  31([x-4]2-[;t;-6]2). 

Divide  through  by  2^'— 9. 


256  UNIVERSITY   ALGEBRA. 

9.   (ia'-xy  =  (a—2xy—x'K 

10.  {x-ay={a-by--{x-by, 

Qx+b      l  +  8.r_l— ^     ^--x 
"•  Sx-lb        15""""    3  ""^~5~* 

If  some  denominators  are  monomials  and  the  other  denominators 
are  not,  it  is  good  practice  to  clear  of  monomial  denominators  first. 
Thus,  multiplying  by  15, 
15(6^+5) 


8a;-15 


--1— 8x=:5-5^+9-3x, 


15(6£+5) 

or  -^ T^— =15, 

8^—15 

whence  6:r4-5=8;c— 15, 

or  ^=10. 

9_^         4  {x-V)Z 

"•      2    ^x-2  2 

10-x    \Z-\-x    7x+26     11+ Ax 


13. 


.;i:+21  21 


3^-4_  7 

14. —r-  =  X''-\-ZX — :-• 

x-\-l  x+1 

X"^    3/    1        1\  23 

^5.  — 


)U-i    3;   ] 


1     6\;ir-l      3/     10(;t:-l) 

T,       1      ,2(^+1)        3^^+l         . 
'•  3;ir— 1"^    x-1        3;t;2-4;c+l       * 

;tr2  — 1  ;»;—l 


19. 


;f— ^     AT— «__   2(^— ^) 


20. 


B(ab-x[a+d'])     (2a  +  d')d'^x^dx       a^d^         ' 
a+b  "^   a{a+by^        a      {a+ty' 

RATIONALIZATION  OF  EQUATIONS. 

379.  If  the  unknown  number  in  an  equation  appears 
under  a  radical  sign,  the  equation  must  first  be  rational- 


SINGLE   EQUATIONS.  257 

ized  before  the  value  of  the  unknown  number  can  be 
found.     This  is  illustrated  by  the  following  examples. 


(1)  Solve  3V4r-8=Vl3jc-3. 

Squaring  both  sides,  9{4:X—S)  =  13x  —  d. 
Transposing  and  uniting,  23x=Qd, 

whence  x=z3. 


(2)  Solve  vJ+9=5Vjt7-3.   _ 

Squaring  both  sides,  x-\-9=:2Dx—30^x-\-d. 

Transposing  rational  terms  to  one  side  and  irrational  to  the  other, 

2ix=30S/x. 
Dividing  by  6  and  squaring,         lQx^=25x. 
Solving  this  quadratic,  "^=16  ^^  0* 

380.  The  most  expeditious  method  for  rationalizing 
any  given  equation  depends  upon  the  peculiar  form  of 
the  equation,  and  can  only  be  determined  by  the  student 
after  a  little  experience  with  this  class  of  equations. 


(1)  Given  V9+I+^=ll.  (1) 

Transposing  everything  but  the  irrational  term  to  the  right-hand 
side  of  the  equation,  we  obtain 

^fdTx=n-x,  (2) 

Squaring  both  sides,  9  +  ^=121-22^+^*,  >  (3) 

and  solving  this  quadratic,  we  find  x=*7  or  16. 

From  (2)  to  (3)  is  a  questionable  derivation;  for  squaring  both  mem- 
bers of  an  equation,  L—R,  we  have  found  (Art.  376)  to  be  equivalent 
to  multiplying  through  by  Z  +  i?,  and  that  the  resulting  equation  is 
equivalent  to  the  two  equations 

\  Z+.?=0  \        . 
Therefore  (3)  is  equivalent  to  the  two  equations 

I     V9+^=ll-^     )  .. 

|V9  +  ;f+ll-x=0  j  ^' 

f  +V9  +  ^+:>r=:ll  )  ,-, 

Hence,  if  we  understand  equation  (1)  to  read 

the  positive  square  root  ^(9  +  x)+5C=ll, 
17— U.  A. 


258  UNIVERSITY   ALGEBRA. 

then  a  new  solution  has  been  introduced  between  (2)  and  (3).     But  if 
we  understand  equation  (1)  to  read 

either  square  root  ^(9+x)4-^=ll. 
then  it  is  equivalent  to  both  the  equations  in  (5),  and  no  solution  has 
been  introduced.   This  is  because  the  introduced  equation  ±Z-|-i?=0 
is  identical  with  the  original  equation  ±L^=.R. 

In  other  cases  the  student  will  always  find  that  rationalization  7nay 
or  may  not  be  considered  a  questionable  derivation  according  as  we  con- 
sider the  radical  signs  to  call  for  a  particular  root  or  any  root  of  the 
expressions  involved. 

(2)  Solve  •  V^+Vx+6=3.  (1) 

Squaring  each  side  of  the  equation,  we  obtain 

5C+2Vx2  +  6x+x  +  6=9.  (2) 

Transposing  all  but  the  surd  to  the  right-hand  side,  this  becomes 

2V:^^Tai=3-2x.  (5) 

Squaring,  we  obtain         4x2H-24jr=9-12:>c4-4^^,  (4) 

or  %—\. 

What  are  the  questionable  steps?     What  is  their  effect  ? 

The  above  solution  is  really  equivalent  to  the  following: 
V^+V;r  4-6=3. 
Transposing  the  3  to  the  left  member, 

v'^J-f-v:^^— 3=0. 

Multiplying  through  by  rationalizing  factor  of  left  member  (Art.  355), 

(\^+VA:-f-6-3)(\^+V.;r-i-6-|-3)(V^-V.r-H6— 3)(V^-V.^+6+3)=0; 
which  reduces  to  4^—1=0,  (5) 

or  x=\. 

The  introduced  equations  are 

\^+V:r  4-6  +  3=0;  V^-Vx+6— 3=0;  V^-V7+6+3=0. 
Here,  then,  is  an  apparent  paradox:  three  solutions  seem  to  have 
been  introduced,  yet  but  one  is  found.  This  can  be  explained  in  the 
following  manner.  If  we  regard  the  radical  signs  as  calling  for  either 
one  of  the  two  roots  of  the  expression  underneath,  then  the  introduced 
equations  are  all  identical  with  the  original  equation,  and  hence  do 
not  give  rise  to  different  solutions.  If  we  restrict  ourselves  to  using 
in  each  case  that  square  root  which  has  the  sign  given  before  the 
radical  sign,  then  none  of  the  introduced  equations  have  any  solution 
whatever,  and  hence  no  solution  is  introduced  in  this  case. 


SINGLE    EQUATIONS.  259 

KXAMPIvKS. 

Solve  each  of  the  following  equations: 
I.  |/;c -4-4=4.  8.  Vx—\/x—5=V3, 


2.  l/2jr-f6=4.  9.  Vx—7=V''x—14:-\-l. 

3.  Vl0xTT^=5.  10.  l/x—7=---']/x-^l—2. 

4.  •2;r+7  =  V5:r— 2.  II.  x=7'-Vx'^—7. 


5.  14  +  ^4^-40=10.  12.  l/^+20-l/:r-l-3=0. 

6.  Vl6x+9=4V4x-S.      13.  T/;i;+34-T/'3;r-2=7. 

7.  V^^T~x=i+Vx.  14.  T/2;r+l-hl/;i;-3=2l/^. 

20;t:  /-7^ ^  18 

15.  -^  — l/lOy— 9=     ,  +9. 


^     x—1       iZ-r+l  i/^4.|/^__3        3 

16.      ,-       =    ^     ^  .  17. 


l/;i;-l       -^—3  '  *  ]/^__l/^^;_3     x—^ 

GRAPHIC  RKPRESENTATION  OF  EXPRESSIONS  AND  EQUA- 
TIONS  OF  THE   FIRST   DEGREE. 

381.  Consider  the  expression  2;t:+3.  If  we  assign 
different  values  to  x  we  will  obtain  different  values  for 
2;r+3.     Thus,  suppose 

^=-4,  -3,  -2,  -1,      0,  +1,  +2,  +3; 

...  2;i;+3=-5,  -3,  -1,  +1,  +3,  +5,  +7,  +9. 
From  this  we  observe  that  as  x  changes  uniformly  from 
—4  to  +3  the  expression  2x+S  changes  uniformly  from 
—5  to  +9.  These  changes  may  be  pictured  to  our  eye 
in  a  simple  way.  On  the  straight  line  of  indefinite 
length,  XX\  select  any  point,  0,  and  lay  off  equal  spaces 
or  units  to  the  right  and  left  of  this  point : 

y,    -4  -3  -2  -1     0   +1  +2  +3  y 

^        I       I       I       I       I       I       I       I -^ 

Fig.  1.— Graphic  representation  of  Algebraic  Scale. 


26o 


UNIVERSITY   ALGEBRA. 


We  may  consider  distances  measured  to  the  right  of  0 
positive  and  distances  measured  to  the  left  of  0  negative. 
In  this  way  there  is  pictured  to  our  eye  the  values  which 
are  assigned  to  x  above.  To  picture  the  values  of  ^x-\-Z 
we  may  draw  at  0  a  perpencicular  line  3  units  long  above 
XX' \  similarly  one  5  units  long 
at  1,  one  7  units  long  at  2,  etc. 
We  may  indicate  the  negative 
values  of  2;t:+3  by  drawing  the 
lines  below  XX' \  thus  at  —2  we 
draw  a  line  1  unit  long  down. 


2.— Graphic  representa- 
tion of  *lx-\-Z. 


382.  If  in  addition  to  the 
values  assigned  to  x  in  the  last  ST'-^jr-r^p^j" 
article,  intermediate  fractional 
values  be  assigned  and  the  cor- 
responding values  of  2;i:-f3  be 
computed  and  placed  in  the  fig- 
ure by  drawing  perpendiculars, 
it  will  be  found  that  the  ends  of 
all  the  perpendiculars  will  be  in  the  straight  line  P^  P^, 
Also  if  the  same  be  done  for  values  greater  than  3  and 
less  than  —4  the  ends  of  these  perpendiculars  will  lie  on 
P^  P^  produced.  For  this  reason  the  line  P^  P^  is  called 
the  Graphic  Representation  of  2;^;+3,  or,  briefly,  the 
Graph  or  Plot  of  2;t:+3. 

In  Fig.  2  we  are  to  understand  that  the  line  /^^  P^  is 
unlimited  in  length. 

BXAMPLKS. 

Represent  graphically  the  following  expressions: 

1.  2;»;-3.  3-  Sx+4:.  5.  -x-2.  7.  4;»;+5. 

2.  x-1.  4.   -'2x+h       6.  -^x+S.         8.  5x--2. 


SINGLE    EQUATIONS. 


261 


Fig.  3. —  Graphic  representa- 
tion of  y=2x-\-Z. 


333.  To  plot  the  equation  y=2x+Z  we  draw  the 
values  of  x  as  above,  and  the  lines  drawn  perpendicular 
to  XX'  in  Fig.  2  are  the  values 
of  y.  The  graph  then  shows 
how  y  changes  as  x  changes. 
We  then  have  the  Graph  of  an 
Equation  instead  of  the  graph  of 
an  expression.  For  this  purpose 
the  method  of  representation  is 
elaborated,  as  the  following  def- 
initions explain. 

384.  The  Axes.  Any  point 
in  a  plane  may  be  located  by  a 
system  of  latitude  and  longitude; 
that  is,  by  giving  its  distance 
from  two  fixed  lines  of  indefinite 
length.  These  lines  are  called  the  Axes  and  are  dis- 
tinguished as  the  :r-axis  and  the  j/-axis.  In  Fig.  3, 
XX'  is  the  Jtr-axis  and  YY'  is  the  j^-axis.  They  are  usu- 
ally, though  not  necessarily,  taken  at  right  angles.  The 
point  of  intersection,  O,  is  called  the  Origin. 

385.  Co-ordinates.  The  distances  of  a  point  from  the 
two  axes  are  called  the  Co-ordinates  of  the  point  The 
co-ordinate  parallel  to  the  ;j;-axis  is  called  the  Abscissa; 
the  co-ordinate  parallel  to  the  j/-axis  is  called  the  Ordi- 
nate. These  displace  the  words  ''longitude"  and  ''lat- 
itude." Abscissas  are  always  represented  by  an  x  and 
ordinates  by  a  y. 

386.  Signs.  Abscissas  when  measured  to  the  right 
are  reckoned  positive,  and  when  measured  to  the  left  are 
negative.  Ordinates  when  measured  up  are  positive^  and 
when  measured  down  are  negative. 


262 


UNIVERSITY    ALGEBRA. 


-2 

-8 

-1 

-5 

0 

—  2 

+  1 

+  1 

4-2 

+4 

+  3 

+  7 

+4 

+  10 

387.  Consider  the  equation  3;*:— 2=^.  By  assigning 
values  to  x  and  finding  the  corresponding  values  of  j/, 
we  may  locate  as  many  of  the 
points  of  the  graph  as  we 
please.   The  values  of  x  and 

y  are  best  given  in 
the  tabular  form  in 
the  margin.  Each 
pair  of  values  on 
the  same  line  lo- 
cates one  point,  as 
when  :r  is  —  2,  jj/  is 
—8.  The  graph  is 
given  in  Fig.  4. 

388.  If  we  wish  to  plot  the  I     ^ 
equation    ax+b=y   we    may        fio.  4.- Graph  oi  3^-2=^. 

proceed  as  follows:  For  the  value  x=0  it  is  plain  y='b\ 
therefore  in  the  figure  (Fig.  4)  we  draw  OB=b,  By 
transposing  the  equation,  we  obtain 

^—=a.  (1) 

X  ^ 

Now  if  we  let  P  be  any  point  on  the  graph,  then  PD=^y 
and  OD=x,  Also  PC-=y^CD=y—OB=y—b,  since  OB 
or  b  is  negative  in  Fig.  4.    Therefore  (1)  becomes 

PC 

CB^""' 
Thus  we  have  shown  that  no  matter  where  the  point 
P  be  taken  on  the  graph,  the  ratio  PC :  CB  equals  a 
fixed  number  a.  Therefore,  by  plane  geometry,  the 
graph  is  a  straight  line.  This  proves  that  every  equation 
of  the  first  degree  in  tivo  variables  represents  a  straight  line. 

389.  Graphic  Representation  of  the  Root.    In  the 

graph  of  a;r+^=^  what  represents  the  root  oi ax-\-b==^^l 


SINGLE    EQUATIONS. 


263 


In  the  equation,  when  j/=0,  x= ;  in  the  graph,  when 

'i/=0,  x=OA,     Therefore  OA^ ,  or  the  distance  from 

the  origin  to  the  point  of  intersection  of  the  x-axis  and  the 
graph  of  ax+b=y  represents  the  root  ofax+d=0. 

KXAMPivES. 

Form  the  graph  of  each  of  the  following  equations  and 
point  out  the  values  of  d,  a,  and  the  root  of  a;r+^=0. 

1.  x-^\=y.     3.  —x-\-\=-y.   5.  Ox-i-4:=y.    7.  —^x+6=y 

2.  2;tr— 3=jK.  4.  ^=y.  6.  0;»;+0=j/.    8.  —2x—A=y 

390.  Graphic  discussion  of  ax-\-d=y.  In  Art.  234 
we  discussed  the  special  forms  of  the  equation  ax +3=0. 
We  may  now  represent  graphically  the  conclusions  there 
deduced. 


'    I.  Suppose  b=0  and  a^O.     Then  in  Fig.  4  OB  [==b'] 

PC 
becomes  zero  and  777,  [_=cl\  is  not  zero.  Hence  the  graph 

takes  the  form  of  Fig.  5  or  Fig.  6  depending  upon 
whether  a  is  positive  or  negative  respectively.  OA,  or 
the  root,  is  zero. 

Jf  b=0  and  a=^0  the  graph  passes  through  the  origin. 


264 


UNIVERSITY   ALGEBRA. 


II.  Suppose  a=0  and  ^=0.  Here  OB  becomes  zero, 
and  y  is  zero  for  all  values  of  x.  Hence  the  graph  coin- 
cides with  the  ;»;-axis.  OA,  or  the  root,  may  be  taken  to 
aiiy  poifit  we  please  on  the  x-axzs,  since  the  graph  meets 
the  X-axis  at  all  points.  This  represents  graphically  the 
indeterminate  value  of  the  root. 

If  a^=0  and  b=0  the  graph  coincides  with  the  x-axzs. 
T  ^ 


Ml 


^ 


Fig.  7. 


PC 
In  this  case  77^  is  zero; 


III.   Suppose  a=0  and  d=t=0. 

that  is,  the  graph  is  parallel  to  the  .r-axis.  It  does  not 
coincide  with  the  .r-axis,  since  d  or  OB^O.  The  graph 
takes  the  form  of  Figs.  7  or  8  depending  upon  whether 
d  is  positive  or  negative. 

r  T 


^ 


X 


^0 


r 

Fig.  9. 


Fig.  to. 


391.  In  Fig.  9  is  represented  the  graph  of  ax+d=y 
when  a  is  not  zero,  but  very  small.  In  Fig.  10  is  repre- 
sented the  graph  ofax-j-d^y  when  a  is  very  large. 

In  Fig.  9  is  ^  represented  to  be  positive  or  negative  ? 


SINGLE    EQUATIONS. 


265 


-3 

-2 
-1 
0 
1 
2 
3 
4 
5 


7 

3 

0 

-2 

-3 

-3 

-2 

0 

3 

7 


GRAPHIC  RKPRKSKNTATION  OF  QUADRATIC  KXPRKSSIONS 
AND   EQUATIONS. 

392.  If  we  wish  to  draw  the  graph  of  any  quadratic 
expression  of  the  form  ax"^  -\-bx-\-c,  or  any  quadratic  equa- 
tion of  the  form  ax'^-\-bx-Yc=y,  we  must  first  obtain  a 

table  of  values  as  indicated  in  the  margin. 
This  set  of  values  came  from  the  equation 
\x'^—^x—2=y.  If  the  points  determined  by 
each  set  of  values  of  x  and  y  in  this  table  be 
plotted  they  will  be  found  to  lie  along  the 
curved  line  represented  in  Fig.  11.  If  this 
table  of  values  be  extended  by  assigning  other 
values  to  x  and  finding  the  corresponding 
values  of  _y,  or  if  any 
number  of  fractional  val- 
ues of  X  be  interpolated 

and  the  intermediate  values  of  y  be 

obtained,  in  either  case  we  would 

find  that  all  the  points  located  by 

the  pairs  of  values  would  lie  along 

the  curve  drawn  in  Fig.  11.     For 

this  reason  the  curve  is  said  to  be 

the  graphic  representation  of  the 

equation  ^x'^—^x—2=y, 

393.  The  curve  represented  in 
the  figure  is  called  the  Parabola. 
Its  properties  need  not  be  investigated  here.  It  is  suffi- 
cient to  notice  that  the  two  roots  of  the  quadratic  equa- 
tion ax'^-{-bx+c=0  are  OA  and  OB  (in  Fig.  11),  and 
that  in  the  case  of  example  7  below  these  two  roots  are 
equal  to  each  other,  and  in  example  8  the  two  roots  are 
imaginary. 


\ 

Y 

\ 

\ 

\ 

Q< .  J 

X 

a\ 

Fig.  11. 

266  UNIVERSITY   ALGEBRA. 

KXAMPLKS. 

Form  a  table  of  values  and  draw  the  graph  for  each  of 
the  following  equations,  pointing  out  the  graphical  rep- 
resentation of  the  roots  of  x^-i-ax+d=0  in  each  case. 

1.  x'^--2xS=y.       4.  x'^=j/.  7.  x'^—6x+9=j^. 

2.  x^—6x—7=jy,       5.  x'^—S^j.       8.  x'^—ex+lS=y. 

3.  x^+Sx+2=y.        6.  x^-i-4:x=y.     9.  jt^+S^j;— 3=j/. 

MISCELIyANEOUS   EQUATIONS. 


Solve  each  of  the  following  equations : 

1.  ;t;t+f=i^:ri 

2.  l/x^  —2  Vx  +  X=0,  Divide  through  by  A 

3.  x+5VW^x=4S. 

Subtracting  37  from  each  side  of  the  equation,  we  obtain 
^-37  +  5V37-x:zr6, 
which  may  be  written     —{S'7—x)-^5^dl-x=6, 

or  (37-x)-5V37-;r=-6. 

Putting  y  for  V37  — ^,  this  becomes 

jk2-5j)/=— 6. 
Solving,  J— 3  or  2. 

That  is,  V37-^=3or2, 

whence  37— ;«r=9  or  4, 

and  5C=28  or  33. 

The  same  example  may  be  treated  by  the  method  of  Art  379. 


/^.  2\/x^  —  6x+2—x^+Sx=Sx—78, 

6.  4:X^—4x+20V2x^—5x+6=6x+6e. 

7.  x-^-'2x-^  =  S.  II.   llQ.;^-4  +  l=:2U-^ 

8.  x»—5x"+4:=0.  12.   l/x-h4x~i=5. 

9.  10;i;^'+.;i:«+24=0.  13.  8j»;*— 8;i;-*'=63. 

10.   a;trF:r+37^=^.  14.   C-^— ^)  —  7 ^,=2. 


SINGLE   EQUATIONS.  267 


15.  Sx'^-Ax+15}/Sx^-4:X-6^42. 

16.  2x^—Sx^-i-xi=0.      Result:  ;r=0  or  1  or  8. 

6^-7     5(.r-l)^  1 
^^'  9;i;+6      12;i:+8      12* 

X     hx'^  —  \hx—%  ^x—^  ___ 
'  2"^     10(;r~3)  5~~ "~   ' 


In  example  20  rationalize  the  denominator  of  the  fraction. 


CHAPTER  XVII. 

SYSTKMS    OF  EQUATIONS. 

394.  The  treatment  of  systems  of  equations  of  the 
first  degree  has  already  been  explained.  See  chapter 
XI.  The  present  chapter  gives  an  account  of  legitimate 
and  questionable  transformations  of  systems  of  equations, 
and  methods  of  solution  of  some  systems  in  which  all  the 
equations  are  not  of  the  first  degree.    See  Arts.  239,  252. 

395.  It  is  assumed  throughout  this  chapter  that  the 
equations  of  all  systems  considered  are  independent  and 
compatible  and  that  the  number  of  equations  of  each  sys- 
tem just  equals  the  number  of  unknown  numbers. 

396.  Equivalence  of  Systems.  One  system  of 
equations  is  said  to  be  Equivalent  to  another  system 
ot  equations  when  any  set  of  values  that  satisfies  the 
first  system  satisfies  the  second  system  also,  and  any  set 
of  values  that  satisfies  the  second  system  satisfies  the 
first  system  also.     Thus  the  system 

One  system  of  equations  is  said  to  be  equivalent  to 
several  systems  of  equations  when  any  set  of  values  that 
satisfies  the  first  system  satisfies  one  of  the  latter  sys- 
tems, and  any  set  of  values  that  satisfies  any  one  of  the 
latter  systems  satisfies  the  first  system.   Thus  the  system 

x'^+y=l'i  \  is  equivalent  to  Jjr— 3=0)        .  J  ^+2=0 
X  +J/=  5  j  the  two  systems  [_>/-2  =  0  j  ^^^  |  j/— 7=0 


SYSTEMS    OF    EQUATIONS. 


269 


397.  Of  course  if  but  one  equation  of  a  system  be 
operated  upon  in  any  manner  during  the  solution,  care 
must  be  taken  that  the  tranformation  be  with  a  due 
regard  to  the  theorems  in  Arts.  370-378.  Obviously  no 
operation  which  it  is  questionable  to  perform  on  an 
equation  standing  alone  can  be  legitimately  performed 
upon  one  belonging  to  a  system.  But  in  addition  to  the 
reductions  which  single  equations  may  undergo,  equa- 
tions of  a  system  permit  of  certain  transformations 
peculiar  to  themselves,  and  it  remains  to  investigate 
the  possible  effect  of  these  on  the  solution  of  the  system. 
The  following  theorems  are  designed  to  point  out  the 
effect  on  the  result  of  the  ordinary  steps  in  the  process 
of  elimination. 


398.  If  from  the  system  of  equations 


we  derive  the  system 


{a) 


ib) 


L„=R„ 

where  all  but  the  second  equation  remain  unchanged,  the 
derivatio7i  is  legitimate  if  T  is  a  kjiown  number  not  zero^ 
but  questio7iable  if  T  is  a  function  of  the  unkjiown  nurn- 
bers,  it  being  indifferent  whether  S  is  a  known  number  or 
a  functio7i  of  the  unknown  ones. 

Write  system  {a)  so  that  it  shall  read 

L,-R,=(i  (1) 

L^-R^=^  (2) 


L„-R„=-Q 


W 


270 


UNIVERSITY    ALGEBRA. 


and  system  (b)  so  that  it  shall  appear 

SiL,-R,)-^TiL,^R,-)^0  (4) 


W 


L„-R„=Q 

Firsts  suppose  T  a  known  number.  Then  any  set  of 
values  that  will  satisfy  {c)  must  make  L^—R^,  L^—R^ 
.  .  .  and  L„—R„  each  zero.  But,  from  the  form  of  (3)  and 
(4),  any  set  that  makes  these  zero  must  satisfy  (d)  also. 
Hence  any  solution  of  (c)  is  a  solution  of  (^). 

It  is  seen  from  (3)  that  any  set  of  values  that  satisfies 
(d)  must  make  L^—R^  zero.  Equation  (4)  will  then 
become 

r(z.2-A)=o.  (5) 

Now,  since  Z*  is  a  known  number  not  zero,  this  cannot 
be  satisfied  unless  L^ — R^  is  zero.  Hence  any  set  of 
values  in  order  to  satisfy  {d)  must  make  L-^ — R^  and 
L^—R^  and  also  .  .  .  L„—R^  each  zero.  But  any  set  of 
values  that  makes  these  zero  will  satisfy  (r).  Therefore 
any  solution  of  {d)  is  a  solution  of  (c). 

Now  we  have  shown,  first,  that  any  set  of  values  that 
will  satisfy  (c)  will  satisfy  (d),  and  second,  that  any  set 
of  values  that  will  satisfy  {d')  will  satisfy  {c).  Hence,  by 
Art.  396,  the  two  systems  are  equivalent. 

Second,  suppose  T  a  function  of  some  of  the  unknown 
numbers.  In  this  case  equation  (5)  may  be  satisfied  by 
any  set  of  values  that  will  satisfy  the  equation  7^=0 
without  requiring  that  L^—R^,  be  zero.  Consequently 
(d)  can  be  satisfied  without  equation  (2)  being  satisfied  ; 
that  is,  without  system  (c)  being  satisfied.  Therefore 
(d)  is  not  equivalent  to  (r),  but  to  the  two  systems 


L2—R2=0 
L„-R„=0 


and 


(L^'-R^=0 


SYSTEMS    OF    EQUATIONS.  2/1 

The  derivation  discussed  in  the  above  theorem  is  the  one  so  fre- 
quently used  in  elimination.     Thus  take  the  system 

2^+7=17  (1)  ) 

5;»:-10y=  5  (2)  j" 

Multiplying  (1)  through  by  5  and  (2)  through  by  2  and  obtaining  a 

new  equation  by  subtracting  the  former  from  the  latter,  the  system 

becomes  2:>c+;/  =  17  (3)  ) 

25y=75  (4)  ] 

We  have  eliminated  x  from  the  second  equation,  and  consequently  y 
is  readily  found  to  equal  3.     From  (3)  x  is  then  found  to  equal  7. 
The  theorem  also  shows  that  it  is  legitimate  to  transform 

•^  ■^^=  y  \  into  \  -^^-^  ^y 

x^  +  (y=:xy  f  ^^^°  ]  6-3x=0 
by  multiplying  the  first  equation  through  by  x  and  subtracting  the 
resulting  equation  from  the  second. 

399.  /i^  ^s  legitimate  to  derive  from  the  system 

i\-k  }  w 

ike  system  L^=R^  \   .,. 

if  T  is  a  known  number  not  zero. 

Rewrite  {a)  and  (U)  so  that  they  shall  read 

^2-^2=0  (2)1     W 

and  Zi^^i=0  (^)lr^^ 

It  is  evident  that  any  set  of  values  which  will  satisfy 
(<:)  will  satisfy  (^);  for  whatever  makes  L^—R^  and 
Lc^—R^  each  zero  will  satisfy  (^).  It  is  seen  from  (3) 
that  any  set  of  values  that  satisfies  (^)  must  make 
Lx—R\  zero.     Equation  (4)  will  then  become 

r(Z2~i?2)=o.  (5) 

Now,  since  T'  is  a  known  number  not  zero,  this  cannot 
be  satisfied  unless  L^—R^  is  zero.  Hence  any  set  of 
values  that  satisfies  {d)  must  make  L^ — R^  and  L^—R,^ 
€ach  zero;  that  is,  must  be  a  solution  of  (r). 


2/2  UNIVERSITY   ALGEBRA. 

Now,  since  any  solution  of  (c)  is  a  solution  of  (^ '  and 
any  solution  of  (d)  is  a  solution  of  (^),  the  two  systems 
are  equivalent. 

As  an  example,  consider  the  system 

^  +7  --=  5  (1)  ) 

;^2+;/8=13  (2)  S 

Multiplying  (2)  through  by  -1  and  adding  to  this  the  square  of  the 
members  of  (1),  we  obtain  the  equivalent  system 

x-\-y=b  (3)  \ 

x^-^2xy+y^-x-^-y^  =  l2  (4)  j" 

that  is,  x+y=5  (5)  ) 

xy=Q  (6)  f 

Squaring  (5)  and  subtracting  4  times  (6)  from  it,  we  obtain  the  equiv- 
alent system  x-\-y=5  (7)  ) 
x'^-2xy+yz=zl                                         (8)  f 
which  is  equivalent  to  the  /w^  systems 

^+^=?  i  and  \  ^^+^=5  , 
that  is,  to  the  solutions     x=3  )        ,  j  x=2 

y=z2  \  ^^    }y=^ 

Note  that  the  solution  of  any  system  of  equations  consists  merely 
of  the  simple  systems  of  linear  equations  (of  the  form  x=a,  y=^, 
etc.)  which  together  are  equivalent  to  the  original  system.  Thus, 
[jr=r3,  y—2]  and  [x=2,  y=S],  which  is  the  solution  of  [x+/=5, 
x'^  -{-y^  =  l3],  are  equivalent  to  the  latter  system.  A  solution  to  a  sys- 
te?n  of  equations  is  another  system  or  several  systems. 

The  above  theorem  evidently  holds  if  the  cube  or  any  other  power 
of  the  members  of  (1)  be  used  instead  of  the  square. 

400.  If  from  a  system  containing  two  unknown  nu7nbers 

we  derive  the  system         L^=^R^  (3)  I    ^a\ 

L,L,=R,R,  (4)  I  W 

the  derivation  is  questionable  if  L^  and  R^  both  i7wolve  un- 
known numbers y  but  legitimate  if  either  is  a  known  number 
not  zero. 

First,  suppose  that  L^  and  R^  each  involve  unknown 
numbers.     Any  value  of  the  unknown  numbers  which 


SYSTEMS    OF    EQUATIONS.  2/3 

will  satisfy  the  equation  Z^=0  must  satisfy  equation 
(4),  since  the  relation  L^—R^  must  hold  if  system  (J?) 
is  to  be  satisfied.  Also  any  value  of  the  unknown  num- 
bers which  will  satisfy  the  equation  R^=0  must  satisfy 
(4),  since  the  relation  L^=R^  must  hold  if  the  system 
is  to  be  satisfied.  Moreover,  au}^  value  of  the  unknown 
numbers  which  will  satisfy  L^—R-i  must  satisfy  (4), 
since  the  relation  L-^=Ri  must  hold. 

Therefore,  from  these  considerations,  it  is  evident  that 
the  system  (J))  is  not  equivalent  to  system  (a),  but  to  the 
three  systems 

Second,  suppose  that  either  L^or  R^\S2,  known  num- 
ber not  zero.  One  of  them,  say  R^,  is  the  known  num- 
ber. Therefore  L^  cannot  be  zero,  since  the  relation 
L^=Ri  must  hold.  Hence  the  introduced  systems  (^j) 
and  (^2)  ^^^  absurdities,  since  they  require  that  L-^  and 
R^  be  zero.  Consequently  the  derivation  is  legitimate, 
since  the  equations  of  the  introduced  systems  are  in- 
compatible. 

As  an  illustration  of  the  theorem,  consider  the  system 
x-4=G-y  I 
2x+y=13      \ 
This  is  satisfied  by  x=3  and  y=7.     Now  form  the  system 
x-4=:C)-y        I 

which  is  satisfied  by  either  of  the  sets  of  values,  x=3,  y=l,  or  ^=4, 
;j/=6.  The  additional  solution  may  be  obtained  from  either  of  the 
systems  x-A^G—y)  x—4^=G-y\ 

x-4=0        i"  G-;/=0        i* 

As  another  example,  consider  the  system 

x-{-2y=l  S 
which  is  satisfied  by  ^=5,  ;'= 1.    From  this  we  may  obtain  the  system 

:tr2 -4/2=21  S 

18— U.  A. 


274  UNIVERSITY   ALGEBRA. 

From  the  first  equation  of  the  system,  we  obtain 

x=3  +  2y. 
Substituting  this  value  for  x  in  the  second  equation,  it  becomes 

9+12>/+4>/2-4)/2r=21; 
whence  J=l- 

Therefore,  from  the  first  equation  of  the  system, 

x=5. 

In  this  case  we  see  that  no  solution  has  been  introduced.     In  fact, 

the  introduced  systems  are  incompatible,  viz. : 

A7-2>/=3  )       x-2y-^  ) 

5C-2y:=0f  7=0  f 

40  It  If  from  the  system 
we  derive  the  sy stern        L^=^R^ 


^2=^2 


}  w 
}  (^) 

the  derivation  is  questionable  if  both  L^  and  R^  involve 
unk?iown  numbers,  but  legitimate  if  either  is  a  k?iown 
number   not  zero. 

First,  suppose  that  L^  and  R^  both  involve  unknown 
numbers.  Then,  by  Art.  400,  if  we  pass  from  {U)  to  {a) 
we  gain  solutions.  Hence  to  pass  from  {a)  to  {U)  we  lose 
those  solutions. 

Second,  suppose  that  either  Z^  or  7?^  is  a  known  num- 
ber. Then,  by  Art.  400,  if  we  pass  from  {U)  to  {a)  no 
solutions  are  gained.  Hence  none  are  lost  if  we  pass 
from  {a)  to  Qf). 

According  to  the  above  theorem  it  is  legitimate  to  divide  one  equa- 
tion by  another,  member  by  member,  if  one  member  is  a  known 
number  not  zero. 

(1)  Take  the  system  x-\-y=i^  \ 

and  derive  the  system  jr+r=5  )      ,., 

x~;/=3  \     (^) 

The  only  set  of  values  which  will  satisfy  {a)  is  j:=4,  y=\.     This 

set  satisfies  {b),  and  no  solution  is  lost.     The  system  {a)  is  equivalent 

to  the  system  {b)  and  to  two  other  systems  (see  Art.  400),  but  the  other 

two  systems  are  incompatible. 


(^) 


SYSTEMS    OF    EQUATIONS.  275 

(2)  As  an  example  of  the  case  in  which  solutions  may  be  lost,  con- 
sider the  system  X  =3—y  I      / \ 

x^  =  C)-y2  \      ^^> 

which  is  satisfied  by  either  of  the  two  sets  x=0,  y=3  and  ^=3,  y=0. 
If  we  divide  the  second  equation  by  the  first,  member  by  member, 
we  pass  to  the  system  x=3—y 

which  is  satisfied  only  by  the  values  x=zS,  y=0. 

(3)  Consider  the  system  x  —2y  =  S 

x^-4y^=21 
x^-2y»=Sd-g 
By  Art.  398  this  is  equivalent  to 

X  -2y  =  3 
x^-4:y^=2\ 
2y^=12-z 
By  the  above  theorem  this  is  equivalent  to 

x-2y=  3 
x  +  2y=  7 
2y^=l2-z 
By  Art.  398  this  is  equivalent  to 

^~^  ^  which,  by  Art.  399,  ^  ^~^ 


y=i  [Whi....     uy    ^IL.     ^VV,^  J 

2y2  =  \2-z)      IS  equivalent  to     ^^     ^ 


10 


UNKAR-QUADRATIC    SYSTEM. 


402.  We  now  take  up  the  solution  of  those  systems 
involving  two  unknown  numbers  which  consist  of  one 
linear  and  one  quadratic  equation.  It  is  convenient  to 
call  this  a  Linear- Quadratic  System.  We  proceed  by 
first  working  the  following  example : 

x''-2y''  =  l  (2)1  W 
From  equation  (1)  the  value  of  x  in  terms  of  y  is  easily 

seen  to  be                        x==5—y.  (3) 

Substituting  this  value  of  x  in  equation  (2),  we  obtain 

(5-J/)2-2j/2  =  l,  (4) 

or                           25-10y-\-y^-2jy^  =  l.  (5) 
Uniting  and  transposing  terms, 

^2  +  i0j=24,  (6) 


2/6  UNIVERSITY   ALGEBRA. 

whence,  solving  this  quadratic,  jk=2  or  —12,  and  from 
equation  (1),  x=S  or  17.  Consequently  there  are  two 
sets  of  values  which  satisfy  (a),  namely: 


y=2]         j/=-12| 


403.  General  System.  The  method  used  may  be 
applied  to  the  solution  of  any  linear-quadratic  system 
containing  two  unknown  numbers.  In  fact,  take  the 
general  case  x-\-ay=b  1   ,^ 

x'^-\-cy'^+dxy+ex+fy=g  ] 

where  a,  b,  c^  d,  e^  f,  and  g  are  supposed  to  stand  for  any 
real  numbers  whatever.  The  value  of  x  in  terms  of  y 
from  the  first  equation  of  the  system  is 

x=b—ay  (1) 

Substituting  this  for  x  in  the  second  equation  of  the 
system  and  combining  those  terms  in  the  second  equa- 
tion which  contain  y^  and  those  which  contain  y  and 
transposing  all  the  known  terms  to  the  right-hand  side, 
we  obtain  the  system 

x=b-ay  (2)1 

{a'^^-c-ad')y'^-\'{bd-1ab-ae-{-f)y^g—b'^'-be    (3)  j 
in  which  (3)  is  a  quadratic  having  y  as  the  only  un- 
known number;  whence  it  can  be  solved.     The  values 
which  may  be  found  from  this  can  be  substituted  in 
equation  (2)  and  the  values  of  x  will  be  determined. 

EXAMPLES. 

Solve  each  of  the  following  systems : 

1.  .;t:2— 3^2^52,         3.  ;r2-jj/2  =  — 9,  5.  ;t:+j^=6, 
2;tr— 3jj/=8.                 x—y=—\,  xy=5. 

2.  3jry-4)/2  =  20,       4.  x''+y^  =  2,  6.  2y'^+xy=14:, 
Sx—iy^lO.               x-i-y=2.  2y+x=7. 

7.  .A;2+3rj/-j>/2=3,         8.  5jr2— 3.ry— 2j/2  =  12, 
Sx-y=h  2x-y=B. 


SYSTEMS    OF    EQUATIONS.  2// 

SYSTEMS   OF  TWO   QUADRATICS. 

404.  If  we  have  a  system  of  two  quadratic  equations 
containing  two  unknown  numbers  and  attempt  to  elimi- 
nate one  of  the  unknown  numbers,  it  will  be  found  in 
general  that  the  resulting  equation  is  of  \h<t  fourth  degree. 

Thus  take  the  system 

We  find  from  the  first  equation  that 

Substituting  this  value  for  jj/  in  the  second  equation, 

or  expanding  and  collecting  terms, 

x^+x^-10x'''-5x-}-16=0. 
Now,  since  we  are  not  yet  familiar  with  the  solution 
of  equations  of  a  degree  higher  than  the  second,  the 
treatment  of  systems  of  two  quadratics  in  general  cannot 
be  taken  up  at  this  place.  But  there  are  two  important 
special  cases  of  systems  of  two  quadratics  whose  treat- 
ment will  involve  no  knowledge  beyond  the  solution  of 
quadratic  equations,  and  these  we  shall  now  consider. 
These  cases  referred  to  are : 

I.  Systems  in  which  the  terms  containing  the  unknown 
numbers  in  both  equations  are  all  of  the  second  degree  with 
respect  to  the  unknown  numbers,  such  as  the  system 

x'^—lxy  =  5) 
3;r2-10j/2  =  35| 

II.  Systems  in  which  both  equations  are  Symmet- 
rical; that  is,  such  that  changing  x  into  y  and  y  into  x 
in  every  term  does  not  alter  the  equation.  The  equa- 
tions of  the  following  system  are  symmetrical: 

xy+x^-y^Z^     j 


278  UNIVERSITY    ALGEBRA. 

But  both  equations  of  the  following  system  are  not  sym- 
metrical: x'^  +y'^  ~\-x+y=  18  | 
x'^—y'^+x—y=  63 

405.  Case  I.  The  following  example  illustrates  the 
method  of  elimination  which  may  be  applied  to  any  ex« 
ample  coming  under  Case  I  above. 

Solve  the  system       x'^—xy=^2  (1)) 

2;t:2+jK2  =  9  (2)1 
Substituting  vx  in  place  of  y,  we  obtain 

x'^'-  vx^^2  '      (3) 

2x^+v'^x^=9  (4) 

2 
From  (3)  we  obtain        ^^—TZT'  (^^) 

9 
From  (4)  we  obtain        ^^  =  0  ,    2  (^) 

2  9 

Whence  f— =2+^  (^> 

Clearing  of  fractions, 

4+2v'^=9-9v  (9) 

or  2v^+9v=5 

Solving  this,  z;=-|-  or  —  5 

Substituting  each  of  these  values  in  turn  for  v  in  (5)  or 

(6),  x=±2ordzll/S 

And  since  j^=z^jr,     y=^l  or  qpfl/S 

Whence  we  have  the  four  solutions  _ 

;i:=2|      x=-21      x=l}/S      I      ^=-11/3  I 
y=li      J/=-lj      j,/=-|i/3i      j|/=|l/3       I 

406.  General  Case.  The  general  system  of  two 
equations  of  the'  above  class  may  be  represented  by 

a^x'^+d^xy+c^y'^=d^  W  1  Ta^ 

a2X^-{-d2xy-}-C2y^=d2  (2)  j   ^  ^ 

Multiplying  (1)  by  ^2  ^^^  (2)  by  d^  and  subtracting, 


SYSTEMS    OF    EQUATIONS.  279 

Dividing  through  hy  y'^,  we  obtain  an  equation  of  the 

^(£)Vi.(£)+c.o. 

From  this  equation  two  values  of  x-r-y  can  be  obtained, 
say  r-^  and  r^.  Then  we  have 

x==7^y 

x=^r^y 
Either  of  these  being  a  linear  equation,  by  combining  (1) 
with  each  in  turn  we  shall  have  two  linear-quadratic  sys- 
tems and  shall  obtain  in  general  four  solutions  to  the 
original  system. 

407.  Case  II.  To  show  that  any  system  of  two  sym- 
metrical quadratics  can  be  solved  we  may  take  the  general 
case  of  this  system,  as  follows : 

x'^-^a^x^-b^xy+a^y+y'^^c^  (1)  | 

x'^^-a^x-^b^xy^a^y-\-y'^=c^  (2)  |  W 

Substitute  u-\-w  for  x  and  u—w  for  j/,  u  and  w  being  two 
new  unknown  numbers.     Then  {a)  becomes 

(2-f^2>2+2a22^  +  (2-^2>'=^2  (4)3   ^^^ 

Multiplying  f3)  by  1—h^  and  (4)  by  l—b^  and  subtracting 
we  obtain  an  equation  of  the  form 

Au'^-^Bu^C,  (5) 

from  which  two  values  of  u  may  be  obtained.     Substi- 
tuting in  (3)  or  (4),  two  values  of  w  may  be  obtained. 
Finally  x  and  y  may  be  determined  from  the  equations 
x=^u+w 

The  above  work  shows  that  Case  II  can  always  be 
solved,  but  we  do  not  pretend  that  the  method  used  is 
always  the  most  economical  one  to  employ.  The  inge- 
nuity of  the  student  will  often  suggest  for  particular 
examples  special  expedients  which  are  preferable  to  the 
general  method. 


28o  UNIVERSITY    ALGEBRA. 

KXA.MPLKS. 

Solve  each  of  the  following  systems  of  equations.. 

x^--2xy=8:  x^—5j/^=:  —  l. 

2,  Axy=4Q,  5.  ^2_3^_^_^2+29=0, 

3.  ^2+y,=  25,  6.  Aix+jrj-^Sxj^O, 
xj/—jy^  =  —4:,  x^+x+y+j/^=20, 

SPECIAL  EXPEDIENTS. 

408.  In  solving  systems  of  equations  above  the  first 
degree  special  expedients  are  more  to  be  sought  for  than 
general  methods.  The  following  solutions  will  tend  to 
point  out  some  of  the  more  common  artifices  made  use  of. 

(1)    Solve  x2+>'2rr20,  (1) 

x-]-y=(j.  (2) 

Squaring  (2),  x^+2xy+y^=d6.                                          (3) 

Subtracting  (1),  2xy=lQ.                                                  (4) 

Subtracting  (4)  from  (1),  x^-2xy+y'i=4:; 

whence  x—y=±2. 

But  from  (2),  x+y=Q. 

Therefore  ^=4  )  ^„  ,  j  x=2 

(2)cSolve  x+y=Q,  (1) 

xy=5.  (2) 

Squaring  (1),  x^  +  2xy  ^y^ =36.  (3) 

Subtracting  4  times  (2)  from  (8), 

x^—2xy+y^z=zl6; 
whence  x  —y=  ±  4. 

But  from  (1),  x-\-y=Q. 

Therefore  ^^f  [  and  j  ;=1 

(3)  Solve  x^-\-y^=Zi,  (1) 

xy=\b.  (2) 

Adding  2  times  (2)  to  (1),     x^i-2xy-^y»=Q4:.  (3) 


SYSTEMS   OF   EQUATIONS.  28 1 

Subtracting  2  times  (2)  from  (1), 

x2-2xy+y^=4.  (4) 

Wnence,  from  (3)  and  (4),        x-{-y=±S, 
x-y—±2. 
Therefore  ^~5  )      x=^  )      5C=  -5  )      ar=-3 ) 

y=^S    y=^S     y=-^S    y=-^o\ 

(4)  Solve  ^8+jj/8=72,  •  (1) 

x+y=:Q.  (2) 

Cubing  (2).  5c8  +  3;r«;/+3xr2+;/8=21C.  (3) 

Subtracting  (1)  and  dividing  by  3, 

xy{x+y)=4S>'^  (4) 

whence,  since  x-\-y=(S,  xy=8.  (5) 

From  (2)  and  (5)  as  in  example  (2)  we  find 
^=4  )  x=.2  ) 
y=2 f     y=i f 

(5)  Solve  x^+y^-12=x+y,  (1) 

xy  +  8=Z{x+y).  (2) 

Adding  2  times  (2)  to  (1),    {x+y)^—5{x+y)=-4:,  (3) 

or  completing  square,       {x+y)^ —5{x-\-y)-\-^£-=^;  (4) 

whence                                         x+y=4:  or  1.  (5) 

Therefore  (2)  becomes               xy=0  or — 6.  (6) 
Solving  (5)  and  (6)  as  in  example  (3),  we  find 

^=0)       x=4  I       x=S      I      x=—2l 

y=4:\     y=os     y=-2S     y=B  S 

(6)  Solve                                  x*+y*=2'72,  (1) 

x-^y=Q.  (2) 

Raising  (2)  to  the  fourth  power, 

x'^-{-4x3y-{-Gx^y»-{-4:Xy^-\-y'i  =  l2d(j,  (3) 

Subtracting  (1)  from  (3), 

^x^y-\-Gx^y^-\-4xy»  =  1024:,  (4) 

Factoring,  xy{2x^-^dxy-\-2y^)=512.  (5) 

Squaring  (2)  and  multiplying  by  2xy, 

xy{2x^-{-4xy  +  2y^)=72xy.  (6) 

Taking  (5)  from  (6),  {xy)^='72xy-512.  (7) 

Solving  this,  xy=Q4:  or  8.  (8) 

Taking  (8)  with  (2)  we  have  two  systems  like  example  (2). 
x-\-y=Ql  x-\-y=G    ) 

xy=8  )  xy=64^  f 

Solving  these,  we  find  

x=4:  I      x=2  I      x=3-fV-55  j_      x=3-\^^  ) 
jK=2f      jj/=4f      ^=3-V-55)      jK=3+V355f 


282 


UNIVERSITY   ALGEBRA. 


KXAMPI^KS. 


Solve  each  of  the  following  systems : 


I.  ;r2+j/2=25, 
:r>/=12. 

3.  2;r2+3j/2  =  e2, 
2^2_3^2=,38. 

;irj/=20. 

5.  ;i:2  +  3xj/-2y  =  26, 

6.  ;i;2-jv2  =  27, 

7.  jr»— j|/3=63, 
^-jV=3. 

8.  Jt2+;rj/+j/2=:9i^ 

9.  ;r2+j/2+^+^_i8, 

10.  x^+y'^+x—y=78y 
xy=^24:. 

11.  42(^2^^/2)^35^^^ 
lbxy=2b20. 

12.  x^— jj/^  =  19(^— -j^), 

jr^— JJ/2=5. 

13.  -^^+y=97, 
^2^y_13, 

14.  j»;3_f.^3  =  i52, 


15-  xy-\-x=20, 
xy—y==12, 

16.  16^2  ==9y, 
2xy—bx-h6y=S3. 

17.  .;i;S+jk3==i33^ 

18.  (^x^+y'^X^-^y')  =  272, 

x'^+y'^+x-\-y=^A2. 

19.  ^2+^JI/=K-^+j/), 

20.  xy — ^2_^2__ — 21^ 
x^+y^+xy=22B. 

21.  xy-hxy'^  =^12, 
x-hxy^=:lS. 

22.  x'^-\-Sxy=o4:, 
xy-i-4y'^  =  115, 

23.  .;r2+xj/+2y2=2, 

2^2_-._^^^^2=2. 

24.  x^+bxy+y'^=51, 
x'^—xy+y^=21. 

25.  ^2+y  =  34^ 
2x^-'Sxyi-2y^  =  23. 

26.  x2+j/2=;r>/4-7, 
;r2— jj/2  =  jt^ — I 

27.  x'^-\-xy=4:, 
x'^-\-2y'^—xy=8. 

28.  2xy+y^  =  16, 
2x^—xy=12. 


SYSTEMS   OF   EQUATIONS. 


283 


30.  ^j+^=10, 
11      9 

32.  ^^^^4-1/^=  12, 
;i:^=1225. 


^^    X    j/""36' 

a      I? 

35.  ^-2y^= 1. 

36.  ^+jj/=^— l/;i;+^, 


37.   (^+j/)(^y"j/*)=13, 
(;t:-j/)(jr*+j|/i)  =  25. 
38.  j/+^  =  3xjj/3',  39.  x{x+y-\-z)=a^, 

2-\-x=2xy2,  y(x-{-y-\-z)=^b'^, 

x-\-y—'^xy2.  z{x-\-y+2)—c'^, 

GRAPHIC   REPRKS:^NTATlON   OF  SYSTEMS   OF  I^INBAR 
EQUATIONS. 

409.  If  we  wish  to  repre- 
sent graphically  the  system  of 
equations  Zx—y=^2 

x~2y=—6 

we  maj^  obtain  a  table  of  val- 
ues from  each  equation,  then 
draw  the  graph  of  each  equa- 
tion on  the  same  axis  of  ref- 
erence, a  shown  in  Fig.  12. 


Fig.  12  —  Graph    of  the    system 


410.  Geometrical  Meaning  of  Solution  of  a  Sys- 
tem. Any  set  of  values  of  x  and  y  that  satisfies  an 
equation  containing  x  and  y  locates  some  point  on  the 


284  UNIVERSITY    ALGEBRA. 

graph  of  that  equation.  Consequently  any  set  of  values 
of  X  and  y  that  satisfies  both  equations  of  a  system  of  two 
equations  contairiing  x  a?id  y  must  locate  some  point  com- 
mon to  the  graphs  of  the  two  equations.  In  other  words, 
the  coordinates  of  a  point  of  intersection  of  two  graphs 
is  a  solution  of  the  equations  of  the  graphs  considered  as 
simultaneous  equations.  Thus  in  Fig.  12  above  the  coor- 
dinates of  P,  the  point  of  intersection  of  the  lines,  are  2 
and  4,  and  ;tr=2,  j>/=4  is  the  solution  to  the  system 
Zx-y=1,  x-2y=6. 

411.  Graphic  Discussion  of  a  System.  In  Art.  252 
we  discussed  the  special  forms  of  the  system 

a^x-i-d^y=c^  (1)  |  .  . 

We  may  now  represent  graphically  the  conclusions 
there  deduced.  For  this  purpose  it  is  convenient  to  have 
(a)  in  one  of  the  following  forms : 


(^)  ^,      .c. 


«2  ^2 


W 


The  equations  in  (^)  are  in  the  form  of  ax+b=y;  there- 
fore, by  Art.  388,  the  coefficients  of  x,  ——^  and  — ,-, 
determine  the  direction  of  the  lines  with  reference  to  OX 
(the  ratio  PC\CB  in  Fig.  4),  and  -^  and  ^^  determine  the 
points  of  intersection  of  the  lines  with  the  j^-axis  (the 
distance  OB  in  Fig.  4).     By  analogy  we  may  note  that 

in  the  equations  in  {c)  the  coefficients  of  jk, and 

b  ^1 

— -,  determine  the  direction  of  the  lines  with  reference 

<^2  c  Co 

to  O  K  and  —  and  —  determine  the  points  of  intersec- 
tion  of  the  lines  with  the  .^-axis. 


SYSTEMS   OF   EQUATIONS. 


285 


I.  Suppose  aiC2—ci2Ci=0  and  a^b<2^—a<2bx'=l^^\  that  is ^ 

suppose  -^=— ^  and  —4^~,     Whence  from  (c)  tlie  lines 

cross  the  ;r-axis  at  the  same  point.     Since  — ^--,  the 

a^     ^2 

directions  of  the  lines  are  not  the  same.     Therefore  we 

have  two  different  lines  intersecting  on  the  :r-axis. 

If  a^c^—ciiC-^^^^  and  a^b^—ct^b-^^O,  the  graph  is  two 
distinct  lines  intersecting  on  the  x- axis.  Ifb^c^ — ^1^2^=^^ 
and  a-^b^  —  d^^i^^i  the  graph  is  two  distinct  lines  inter- 
seeling  on  the  y-axis. 


II.  Suppose  a^c<2, — <3J2^i=0  and  a^b2 — ^2^i=0;  that  is 

suppose  — ^=— ^  and  -^=-^.     Then  from  (c)  we  see  that 
a-^     a^  a^     a^ 

the  lines  have  a  common  point  on  the  ;r-axis  and  also 
have  the  same  direction.  Therefore  the  graph  is,  two 
coincident   straight   lines.     As   these  lines  intersect  at 


^ 


U X 


SL 


Fig.  13. 


r 

Fig.  14. 


every  point,  we  see  the  appropriateness  of  calling  the 
the  solution  indeterminate.  Since  the  graphs  of  the  sep- 
arate equations  are  coincident,  it  shows  that  the  given 
system  is  really  equivalent  to  but  one  equation  contain- 
ing two  unknown  numbers.  The  graph  is  represented 
hi  Fig.  14.  If  <52^i —^1^2—0  ^"^^  a ^b 2,— a 2b  1=^0 ^  the 
graph  takes  the  form  in  Fig.  13. 


286 


UNIVERSITY   ALGEBRA. 


If  a  ^c  2— (12^1=0  and  a  ^5  2— a  2^  1=0,  the  graph  consists 
of  two  coincident  straight  lines.  If  ^2^i"~^i^2=0  and 
^ii^2"~^2^i~^>  ^^^  ^^^^  ^^  true, 

III.  Suppose  a^C2—a2C-^^0  and  ai^2~'^2^i=0;  that 


is,  suppose 


—  and  -1=-^.    Then  from  (f)  we  observe 


that  the  lines  intersect  the  jc-axis 
at  different  points  and  that  the 
directions  of  the  lines  are  the  same; 
whence  the  graph  consists  of  two 
parallel  lines,  as  in  Fig.  15.  We 
notice  that  the  lines  of  the  system 
do  not  intersect,  and  consequently 
there  is  no  finite  solution  to  the 
system.     The  equations  are  said  to 

be  incompatible,  but  the  lines  are  said  to  be  parallel 
If  a^C2—a2C^^^  and  a^b2—a2b^=^0y  the  graph  consists 

of  two  parallel  lines. 

KXAMPLKS. 

Draw  the  graph  of  each  of  the  following  systems  and 
point  out  the  graphical  representation  of  the  solution  to 
the  system : 

4.  %x^-Zy=Z, 
12jt:+9j/=3. 

5.  5;r— 4j/=6, 
8:i;=7r. 

6.  hx—y^^Xhy 

PROBI.KMS. 


1.  x-hy=6y 
x-y=2. 

2.  7x-i-4j/=l, 
9^ -1-4;/=  3. 

3.  2;t:+jK=ll, 


7.  Sx+Sy=6, 
Sx+Sy=16. 

8.  x+^y=l 
4:x+10y=15. 

g.  Sx'-2y=0, 
2xSy=d, 


I.  What  two  numbers  are  those  whose  product  is  24 
and  whose  sum  added  to  the  sum  of  their  squares  is  62  ? 


I 


SYSTEMS    OF    EQUATIONS.  28/ 

2..  A  number  consists  of  two  digits  whose  sum  is  15. 
If  31  be  added  to  the  product  of  the  digits,  the  digits  will 
he  in  the  reverse  order.    What  is  the  number? 

3.  The  sum  of  the  squares  of  two  numbers  is  410.  If 
we  diminish  the  greater  by  4  and  increase  the  lesser  by 
4  the  sum  of  the  squares  of  the  two  results  is  394.  What 
are  the  two  numbers? 

4.  Find  four  consecutive  integers  such  that  the  product 
of  the  first  two  shall  be  a  number  that  has  the  other  two 
for  digits. 

5.  The  hypothenuse  of  a  right-angled  triangle  is  10 
feet  and  its  area  is  24  square  feet.     Find  the  sides  of  the 

triangle. 

6.  Find  the  sides  of  a  right-angled  triangle ;  given  its 
hypothenuse  equal  to  h  and  its  area  equal  to  a. 

7.  The  perimeter  of  a  rectangle  is  16  feet  and  its  ^rea 
is  15  square  feet.     Find  the  dimensions  of  the  rectangle. 

8.  Find  the  dimensions  of  a  rectangle ;  given  that  its 
perimeter  is  p  feet  and  its  area  a  square  feet. 

g.  Show  that  if  an  isosceles  triangle,  the  sum  of  whose 
sides  is  2^^,  be  inscribed  in  a  circle  of  radius  a,  one  of  the 
two  equal  sides  of  the  triangle  must  equal 

l/3«2±al/9^2_2^^ 

10.  Find  the  side  of  an  equilateral  triangle,  knowing 
that  a  side  exceeds  the  altitude  by  d  feet. 

11.  A  and  B  attempt  the  same  quadratic  equation.  A 
after  reducing  has  only  a  mistake  in  the  absolute  term 
and  finds  for  roots  +8  and  -f  2 ;  B  after  reducing  has  only 
a  mistake  in  the  coefficient  oi  x  and  finds  for  roots  —9 
and  —1.     Find  the  roots  of  the  correct  equation. 


288  UNIVERSITY   ALGEBRA. 

12.  A  man  bought  some  horses  for  $1250.  If  he  had 
bought  3  more  and  paid  $25  less  for  each  horse,  they 
would  have  cost  him  $1300.  How  many  horses  did  he 
buy,  and  at  what  price  ? 

13.  If  a  carriage  wheel  14|-  feet  in  circumference  take 
one  second  more  to  revolve,  the  rate  of  the  carriage  per 
hour  will  be  2|-  miles  less.  How  fast  is  the  carriage 
traveling  ? 

14.  A  sets  off  from  London  to  York,  and  B  at  the  same 
time  from  York  to  London,  and  each  travels  uniformly. 
A  reaches  York  16  hours  and  B  reaches  London  36  hours 
after  they  have  met  on  the  road.  Find  the  time  in  which 
each  has  performed  the  journey. 

15.  A  man  arrives  at  the  railway  station  nearest  his 
home  \\  hours  before  the  time  at  which  he  has  ordered 
his  carriage  to  meet  him.  He  sets  out  at  once  to  walk 
at  the  rate  of  4  miles  per  hour,  and  meeting  his  carriage 
when  it  had  traveled  8  miles,  reaches  home  one  hour 
earlier  than  he  had  originally  expected.  How  far  is  his 
home  from  the  station,  and  at  what  rate  was  his  carriage 
driven  ? 


I 


CHAPTER  XVIII. 

THEORY  OF  I.IMITS. 

412.  Thus  far  in  the  sudy  of  Algebra  the  numbers  we 
have  used  have  always  preserved  the  same  value  through- 
out the  same  problem.  Letters  have  been  used  to  repre- 
sent numbers.  A  letter  may  have  stood  for  one  number 
in  one  problem  and  another  number  in  another  problem, 
but  any  letter  has  always  preserved  the  same  value  in  the 
same  problem.  There  are,  however,  many  cases  in  Alge- 
bra in  which  it  is  desirable  to  consider  expressions  which 
change  in  value  in  the  same  problem,  and  thus  we  are 
led  to  consider  two  kinds  of  number  defined  in  the  next 
article. 

413.  Constants  and  Variables.  When  an  expression 
preserves  its  value  unchanged  in  the  same  discussion,,  it 
is  called  a  Constant ;  but  when  under  the  conditions  of 
the  problem  an  expression  may  assume  an  indefinite  num- 
ber of  values,  it  is  called  a  Variable. 

Constants  are  usually  represented  by  the  first  or  inter- 
mediate letters  of  the  alphabet  and  variables  by  the  last 
letters. 

The  notation  by  which  we  distinguish  between  constants  and  var- 
iables is  the  same  as  that  by  which  we  distinguish  between  known 
and  unknown  numbers,  but  it  must  not  be  thought  that  any  analogy 
is  intended  to  be  pointed  out  by  this  fact.  When  we  are  discussing  a 
problem  in  which  both  constants  and  variables  appear  we  usually  do 
not  care  whether  the  constants  are  known  or  unknown. 

414.  Continuous  and  Discontinuous  Variables. 
When  a  variable  in  passing  from  one  value  to  another 

19— u.  A. 


290  UNIVERSITY   ALGEBRA. 

passes  through  all  intermediate  values,  it  is  called  a 
Continuous  variable ;  when  it  does  not  pass  through  all 
intermediate  values,  it  is  called  a  Discontinuous  var- 
iable. 

415.  Limit  of  a  Variable.  When  a  variable  changes 
in  value  by  approaching  nearer  and  nearer  some  constant 
which  it  can  never  equal,  yet  from  which  it  can  be  made 
to  differ  by  an  amount  as  small  as  we  please,  this  con- 
stant is  called  the  Limit  of  the  variable. 

416.  Illustrations.  If  a  point  move  along  a  line  ^^, 
starting  at  A  and  moving  in  such  a  way  that  the  first 
second  the  point  moves  one-half  the  distance  from  A  to 
B,  the  second  second  one-half  the  remaining  distance, 
the  third  second  one-half  the  distance  which  still  re- 
mains, and  so  on ;  then  the  distance  from  A  to  the  mov- 
ing point  is  a  variable  whose  limit  is  the  distance  AB, 

^ ^ : ^ ^ 

For,  no  matter  how  long  the  point  has  been  moving, 
there  is  still  some  distance  remaining  between  it  and  the 
point  B,  so  that  the  distance  from  A  to  the  moving  point 
can  never  equal  AB\  but  as  the  moving  point  can  be 
brought  as  near  as  we  please  to  B,  its  distance  from  A 
can  be  made  to  differ  from  the  distance  AB  hy  stn  amount 
as  small  as  we  please. 

Thus  we  see  that  the  distance  from  A  to  the  moving 
point  fulfills  all  the  requirements  of  the  definition  of  a 
variable,  and  the  distance  AB  all  the  requirements  of  the 
definition  of  a  limit. 

The  student  must  note  that  it  is  not  the  pozn^  B  that 
is  the  limit  of  the  moving  point,  although  the  moving 
point  approtaches  the  point  B ;  but  it  is  the  dista^ice  AB 
that  is  the  limit  of  the  distance  from  A  to  the  moving 
point. 


\ 


THEORY    OF    LIMITS.  29I 

If  we  call  the  distance  the  point  moves  the  first  second 
1  (then  of  course  the  whole  distance  AB  would  be  2), 
the  distance  traversed  the  second  second  would  be  \,  that 
traversed  the  third  second  would  be  ^,  and  so  on,  and  the 
entire  distance  from  A  to  the  moving  point  in  n  seconds 
would  be  the  sum  of  n  terms  of  the  series 
1     i    1     1.    _i 

-^J    2>    T'    8'    i^»  •  •  • 

Now  it  is  sure  that  the  more  terms  of  this  series  that 
are  taken  the  less  does  the  sum  differ  from  2;  but  the 
sum  can  never  equal  2.  Hence  we  say  that  the  limit  of 
the  sum  of  the  series  l+i+i4-i+YV+-  •  •  as  the  num- 
ber of  terms  is  indefinitely  increased  is  2. 

Again,  consider  any  regular  polygon  inscribed  in  a 
circle.  Join  the  vertices  with  the  middle  points  of  the 
arcs  subtending  the  sides,  thus  forming  another  regular 
inscribed  polygon  of  double  the  number  of  sides.  From 
this  polygon  form  another  of  double  its  number  of  sides, 
and  so  on  Now  the  polygon  is  always  within  the  circle, 
and  hence  the  area  of  the  polygon  can  never  equal  the 
area  of  the  circle;  but  as  the  process  of  doubling  the 
number  of  sides  is  continued  the  less  does  the  area  of 
the  polygon  differ  from  the  area  of  the  circle.  Hence  we 
say  that  the  limit  of  the  area  of  the  polygon  is  the  area 
of  the  circle. 

Again,  as  a  straight  line  is  the  shortest  distance  be- 
tween two  points,  any  side  of  the  inscribed  polygon  is 
less  than  the  subtended  arc ;  hence  the  sum  of  all  the 
sides,  or  the  perimeter,  of  the  polygon  is  less  than  the 
sum  of  all  the  subtended  arcs,  or  the  circumference,  of 
the  circle ;  or  in  other  words,  the  perimeter  of  the  poly- 
gon can  never  equal  the  circumference  of  the  circle.  But 
as  the  process  of  doubling  the  number  of  sides  is  con- 
tinued the  perimeter  of  the  polygon  differs  less  and  less 


292  UNIVERSITY   ALGEBRA. 

from  the  ciecumference  of  the  circle ;  hence  the  circum- 
ference of  the  circle  is  the  limit  of  the  perimeter  of  the 
inscribed  polygon. 

417.  The  student  should  not  infer  from  what  has  been 
said  that  all  variables  have  limits.  In  fact  the  truth  is 
quite  the  contrary,  for  most  variables  do  not  have  limits. 
Thus  in  the  illustration  of  the  moving  point  given  above 
the  variable  does  not  have  a  limit  if  we  suppose  the  point 
to  move  at  a  uniform  rate.  For,  if  the  velocity  is  uni- 
form, it  is  a  mere  question  of  time  until  the  moving  point 
passes  B  or,  in  fact,  any  other  point  to  the  right  of  B, 
however  remote.  Much  more  would  this  be  true  if  the 
point  moved  with  increasing  instead  of  uniform  velocity. 

Again,  consider  the  fraction .     If  x  be  supposed 

to  change  in  value,  the  value  of  the  fraction  changes  and 
is  itself  a  variable.  Now  suppose  x  to  decrease  in  value. 
It  is  plain  that  the  value  of  the  fraction  increases  without 
limit  as  x  decreases.  In  other  words,  the  value  of  the 
fraction  can  be  made  as  large  as  we  please  by  taking  x 
small  enough.  Hence  as  x  decreases  the  value  of  the 
fraction  increases, 

418.  It  follows  immediately  from  the  definition  of  a 
variable  that  the  difference  between  a  variable, and  its  limit 
is  a  variable  whose  limit  is  zero.  For  if  ;i:  be  a  variable 
whose  limit  is  a,  then  x  may  be  made  to  differ  from  a  by 
an  amount  as  small  as  we  please;  hence  a-^x  may  be 
made  as  small  as  we  please.  Yet  as  x  can  never  equal  a^ 
a—x  can  never  equal  zero ;  hence  a—x  is  a  variable  whose 
limit  is  zero. 


THEORY   OF   LIMITS.  293 

419.  Infinitesimal  and  Infinite.  A  variable  which 
approaches  zero  as  a  limit  is  called  an  Infinitesimal. 
The  difference  between  a  variable  and  its  limit  is  there- 
fore an  infinitesimal. 

When  a  variable  increases  without  limit  in  such  a 
manner  that  it  may  become  and  remain  larger  than  any 
assigned  number,  however  large,  the  variable  is  said  to 
be  Infinite  and  is  called  Infinity  and  is  often  represented 
by  the  symbol  00. 

It  is  to  be  noticed  that  the  words  infinite  and  infinites- 
imal do  not  refer  to  definite  magnitudes  at  all,  but  each 
refers  to  something  which  is  essentially  variable.  These 
two  words  are  introduced  in  order  that  we  may  have 
names  to  designate  two  particular  kinds  of  variables 
which  play  an  important  part  in  mathematics.  A  num- 
ber which  is  neither  infinite  nor  infinitesimal  is  called 
Finite. 

THEOREMS  ON   I^IMITS. 

420.  Theorem  I.  If  two  variables  are  continually 
equal  and  each  approaches  a  limit,  the  limits  are  equal. 

Let  X  and  y  be  the  variables  and  let  limit  x=-a  and 
limit  y=^b.  We  are  to  prove  that  a^=^b.  \i a  and  b  are 
not  equal,  suppose  a  greater  than  b  and  let  a—b^d.  Let 
a--x=u  and  b—y=v',  then  a=x+u  and'^=j/-f  z/,  and 
a-^b=^d  becomes  by  substitution 

{x-]ru)  —  {y+v)=d, 
or  (x—y)-\-(u—v)=^d. 

Since  limit  ^=«,  .*.  limit  u=0]  and  since  limit  y=b, 
.'.  limit  v=0.  Thus  we  see  that  u  and  v  are  each  var- 
iables which  may  be  made  as  small  as  we  please,  and 
hence  the  difference  u—v  may  be  made  as  small  as  we 
please,  and  hence  may  be  made  less  than  d.  Therefore 
x—y  equals  some  number;  /.  e, ,  x  and  y  differ,  which  is 


294  UNIVERSITY   ALGEBRA. 

contrary  to  the  hypothesis.  Hence  a  cannot  be  greater 
than  b,  and  in  the  same  way  it  may  be  shown  that  b  can- 
not be  greater  than  a.     Therefore  a=b. 

421.  Theorem  II,  The  limit  of  the  sunt  of  a  constant 
and  a  variable  equals  the  sum  of  the  constant  and  the  limit 
of  the  variable. 

Represent  the  variable  by  x  and  its  limit  by  a.  There- 
fore limit  x=^a.  Let  c  be  any  constant.  We  are  to  prove 
that  limit  (c+x')=c-i-a.     lyct 

a—x=u;  (1) 

.-.  x=a~u;  (2) 

.'.  c+x=c+a—u,  (3) 

Now,  since  x  may  be  taken  so  near  a  as  to  differ  from  it 
by  an  amount  as  small  as  we  please,  therefore  by  equa- 
tion (1)  u  may  be  made  as  small  as  we  please,  therefore 
c+a—u  may  be  made  to  differ  from  c+a  by  an  amount 
as  small  as  we  please.     Therefore 

limit  (^c+a—u')=c+a. 
From  (3),  limit  (c+x)=c+a. 

422.  Theorem  III.  The  limit  of  the  variable  sum  of 
a  limited  number  of  variables  equals  the  sum  of  their  sep- 
arate  limits. 

In  this  theorem  we  speak  of  the  variable  sum  of  a  lim- 
ited number  of  variables  because  it  is  possible  for  two  or 
more  variables  to  be  so  related  that  when  added  together 
the  sum  is  constant,  and  as  it  is  only  variables  which 
have  limits,  it  is  necessary  that  the  sum  of  the  variables 
considered  should  itself  be  variable.  Again,  we  speak 
of  a  limited  number  of  variables  because,  as  will  be  seen 
from  a  single  illustration  given  at  the  end  of  the  proof, 
the  theorem  is  not  necessarily  true  for  an  unlimited  num- 
ber of  variables. 


THEORY    OF    LIMITS.  295 

Let  the  variables  be  x,  y,  z,  etc.,  and  let  limit  ;»;=«, 
limit  j/=/^,  limit  z=c,  etc.;  we  are  to  prove 

limit  {x+y-\-2-^.  .  .)  =  <2  +  ^H-^+.  . . 
Let  a — x^=^ic  .'.  x=a — u; 

d—y=^v  .'.  y=d — v; 

C — Z=W  .'.    2=C — W'y 

etc.  etc. 

Then  (x+y+2+,  .  .)  =  (a  +  ^+^+.  .  .)  —  (u-^v+w-\- .  .  .) 
Suppose  u  to  be  numerically  the  greatest  of  the  numbers 
u,  V,  w,  etc.,  and  suppose  that  there  are  n  of  these  num- 
bers. Now,  since  x  may  be  taken  so  near  a  as  to  differ 
from  it  by  an  amount  as  small  as  we  please,  it  follows 
that  u  may  be  made  as  small  as  we  please.  Therefore 
X  may  be  taken  so  that  u  will  be  smaller  than  any  fixed 
number  that  may  be  named.  Then  letting  k  stand  for 
some  fixed  number,  no  matter  how  small,  x  may  be  taken 

k 
so  that[]u<C— ornu<Ck.  Thenu+v+'w+<inu. ,  .  (since /^ 

is  the  largest  of  the  numbers  u,  v,w,.. .).  Hence,  however 
small  k  may  be,  u  +  v+w-}-.  .  .<k;  that  is,  x-\-y-\-z-^.  .  . 
may  be  made  to  differ  from  a+b -{-€+. .  .  by  an  amount 
as  small  as  we  please.     Hence 

limit  {x'\-y-{-2+,  .  .')=a  +  b+c+,  . . 
This  theorem  is  not  necessarily  true  when  the  num- 
ber of  variables  is  unlimited.     For  example,  the  sum 

— I 1 \-.  .  .  to  jr  terms  is  evidently  unity,  however 

XXX 

great  x  may  be,  but  as  x  increases  without  limit  plainly 
the  limit  of  each  term  is  zero,  and  the  sum  of  the  limits 
is  the  sum  of  a  number  of  zeros. 

423.  Theorem  IV.  The  limit  of  the  product  of  a  con- 
stant and  a  variable  equals  the  constant  multiplied  by  the 
limit  of  the  variable. 


296  UNIVERSITY   ALGEBRA. 

Let  xhe  a,  variable  and  a  its  limit.  We  are  to  prove 
limit  nx=na.     I^et  a—x=u;  then  x  may  be  taken  so 

k 
near  to  a  as  to  make  u<i—  or  nu<Jz,  however  small  k 

n 

may  be.  But  since  a—x=u  .*.  na—nx=-nu\  hence  nx 
may  be  made  to  differ  from  na  by  an  amount  as  small  as 
we  please.  But  as  x  can  never  equal  a^  therefore  nx  can 
never  equal  na.     Hence  limit  nx=7ia. 

424.  Theorem  V.  The  limit  of  the  variable  product  oj- 
two  variables,  each  of  which  approaches  a  limit,  equals  the 
product  of  their  limits. 

With  the  same  notation  as  in  Art.  420  we  are  to  prove 
that  limit  xy=^ab, 

xy=(a — u){b — v)=ab — av — bu-\-uv=ab — {av-Vbu—uv), 
Now  since  limit  z;=0,  therefore  limit  az;=0;  and  since 
limit  2^=0,  therefore  limit  bu=0;  and  since  u  and  v  are 
each  as  small  as  we  please  and  the  product  smaller  than 
either,  limit  uv=0.  And  since  the  limit  of  each  term  of 
av+bu—uv  is  zero,  the  limit  of  the  algebraic  sum  of  all 
three  terms  is  zero.  Hence  xy  may  be  made  to  differ 
from  ab  by  an  amount  as  small  as  we  please ;  hence 
limit  xy=ab. 

425.  Theorem  VI.  The  limit  of  the  variable  product 
of  any  number  of  variables,  each  of  which  approaches  a 
limit,  equals  the  product  of  their  limits. 

With  the  same  notation  as  before  we  are  to  prove 
limit  {xyz  .  .  .)=abc .  .  . 
We  have  already  proved  that  limit  xy=ab,  and  we  may 
consider  xy  a  single  variable  and  ab  its  limit ;  then  by 
the  last  article,         limit  (xy.  2)=ab.c, 
or  limit  xyz=abCy 

and  now  xyz  may  be  considered  a  single  variable  and  abc 
its  limit  and  a  repetition  of  the  application  of  the  theorem 


THEORY    OF    LIMITS.  297 

of  the  last  article  shows  that  the  limit  of  the  product  of 
four  variables  equals  the  products  of  their  limits.  As  this 
reasoning  can  be  carried  to  any  extent  desired,  the 
theorem  is  evidently  true. 

426.  Theorem  VII.  The  limit  of  the  variable  quotieyii 
of  two  variables^  each  of  which  approaches  a  limit,  equals 
the  quotient  of  their  limits,  provided  the  limit  of  the  divisor 
be  not  zero. 

With  the  same  notation  as  before  we  are  to  prove  that 

..    .X     a 
limit  — =^- 

X  y    ^ 

Let  —=^q\  then  x=qy.  We  know  that  limit  ;i:=  limit  qy 
= limit  a .  limit  r;  therefore  limit  0=^- — r: — =-t-     Hence 

limit  — =T. 
y     b 

427.  Theorem  VIII.  The  limit  of  the  reciprocal  of  a 
variable  equals  the  reciprocal  of  the  limit  of  the  variable. 

With  the  same  notation  as  before  we  are  to  prove  that 

limit  f-V- 
\x)     a 

1      X  /1\  X 

Since ^>  therefore  limit  {  —  )=  limit —^-    But  by  Art. 

X    x^  \x)  x^ 

426,  limit  f  4)=-9  or  -•     Hence  limit  ("-)=-• 
\x^)     a^      a  \xj     a 

This  theorem  is  not  proved  when  a  is  zero,  for  we  do 

not  yet  know  what  is  meant  by  -J-.     This  will  be  taken 

up  presently. 

428.  Theorem  IX.  The  limit  of  a  power  of  a  variable 
equals  that  power  of  the  limit  of  the  variable. 

With  the  same  notation  as  before  we  are  to  prove  that 
limit  x^'^^a'',  n  being  either  positive  or  negative,  inte- 
gral or  fractional. 


298  UNIVERSITY    ALGEBRA. 

First.  When  ;^  is  a  positive  integer.     By  Art.  425, 

limit  {xxx  .  .  .')  =  aaa .  .  . 

or  limit  x''=a''. 

i> 
Second.  When  ;2  is  a  positive  fraction,  say  — •     I^t 

^^=jv;  (1) 

then  x-=y\  (2) 

hence  by  Art.  420,     limit  ;r=limit^.  (3) 
If  we  represent  limit  jv  by  b, 
then  by  first  case  of  this  article. 

limit  JV*'=^^•  (4) 

hence  from  (3)  and  (4),      a=b''.  (5) 

From  (1),  .^l=y;  (6) 

p 
hence  by  Art  420,      limit  ;i:^= limit  y^=b^,  (7) 

Bat  from  (5),  b^=a^)  (8) 

hence  limit  x^—ai,  (9) 

Third.  When  ;^  is  a  negative  number,  either  integral 

or  fractional,  say  n=^—s.     Since  ^"^=-3.,  therefore 

limit  :i:~^=  limit  — =r — -^ — ."=—,= ^~'\ 
x^     limit  xf     a" 

'  .*.  limit  x~^=a~\ 

INDKTKRMINATE    FORMS. 

429.  Certain  expressions  occur  in  Algebra  which  for 
particular  values  of  one  or  more  of  the  letters  assume  an 
indeterminate  form.  For  example,  an  expression  may 
assume  the  form  ^  and  this  result  may  be  placed  equal  to 
any  number  whatever  and  the  result  satisfies  the  test  of 
division,  viz.:  the  quotient  multiplied  by  the  divisor 
equals  the  dividend;  e.g.,  ^=6  because  6x0=0.  Of 
course  any  other  number  would  answer  as  well  as  6.  We 
shall  presently  consider  some  of  these  indeterminate 
forms. 


THEORY    OF    LIMITS.  299 

430,  From  tlie  properties  of  fractions  we  know  that  if 
the  denominator  is  a  fixed  finite  number  and  the  numer- 
ator approaches  zero  the  value  of  the  fraction  itself  ap- 
proaches zero,  and  if  the  numerator  increases  without 
limit  the  value  of  the  fraction  itself  increases  without 
limit.  In  other  words,  if  the  denominator  of  a  fraction 
is  a  fixed  finite  number  the  value  of  the  fraction  will  be- 
come infinitesimal  when  the  numerator  becomes  infinites- 
imal, and  infinite  when  the  numerator  becomes  infinite. 
Again,  if  the  numerator  is  a  fixed  finite  number  and  the 
denominator  approaches  zero  the  value  of  the  fraction 
itself  increases  without  limit,  and  if  the  denominator  in- 
creases without  limit  the  value  of  the  fraction  approaches 
zero.  In  other  words,  if  the  numerator  of  a  fraction  is  a 
fixed  finite  number  the  value  of  the  fraction  will  become 
infinite  when  the  denominator  becomes  infinitesimal,  and 
infinitesimal  when  the  denominator  becomes  infinite. 

431.  The  expression  ■§■  considered  by  itself  is  abso- 
lutely void  of  meaning,  and  no  meaning  can  be  assigned 
to  it  until  we  know  how  the  expression  originated.  It 
must  be  remembered,  however,  that  an  expression  pre- 
sents itself  in  this  form  because  of  certain  values  being 
given  to  one  or  more  of  the  letters  in  the  expression. 

For  example,  the  fraction  — ^-; 77-  when  jr=2  assumes 

^  ^  x^-]-x—b 

^  the  form  ■^.     Therefore  in  this  case  we  are  dealing  with 

\'  x'^  —  Qx-\-S 

Is  the  value  of  the  fraction  — ^— 77-  when  ;r=2.     By 

Uhe  value  of  this  fraction  when  x=2  we  mean  the  /imzf 
which  this  fraction  approaches  as  x  approaches  2,  and  to 
find  this  limit  is  the  problem  before  us. 


300  UNIVERSITY   ALGEBRA. 

x^ 6;t;+8 

432.  I^et  us  now  find  the  limit  of  — tt-, tt  as  x 

x^-\-x—o 

approaches  2.    To  express  this  limit  we  use  the  notation 

limit  — K-, ^.     This  is  read  the  limit  of  — ^— j,- 

^Q  x^+x  —  o  x^-\-x—Q 

as  X  approaches  2,  the  symbol  ^  standing  for  approaches. 

^    ,  ;r2-6^+8     (^— 2)  (jr--4)     x—A 

We  know     _^^_^=^-_^-^=— -. 

Therefore  by  theorem  I, 

limit  — 2~i ^=limit  — -5- 

r    >£--^__^ 
But  plainly,  ™  x+^~     b 

Therefore  ^^'^  ^2+^__e  --5' 

KXAMPI.KS. 

By  a  method  similar  to  that  illustrated  above  find  the 
limit  in  each  of  the  following  examples : 

,  limit  r^^±^:^i  ^  limit  rc^+^)^-^n 

^  limit  r_^f_i       -  limit  r^!+i"|  . 

limit  r^i=j^"|      6    Prove  li^it  r^^—l=^^''"^ 

433.  Sometimes  we  have  to  deal  with  the  product 
of  two  expressions,  one  of  which  is  infinite  and  the 
other  infinitesimal :  such  an  expression,  for  example,  as 

Ix^—xi T )  when  x  approaches  zero.     Here  the 

first  factor  approaches  zero  and  the  second  factor  in- 
creases without  limit  as  x  approaches  zero.  When  x—^ 
this  expression  is  absolutely  void  of  nllaning  until  a 


THEORY   OF   LIMITS.  3OI 

meaning  is  assigned  to  it.  By  the  value  of  this  expres- 
sion when  x=0  we  mean  ihe  limit  which  this  expression 
approaches  as  x  approaches  zero.  To  find  this  limit  is  the 
problem  before  us. 

434.  Let  us  now  find  the  limit  of  {-^^  ""-^jf— tt ) 

as  X  approaches  zero.     We  have  by  succesive  reductions 

Therefore  by  theorem  I, 

:;ToC(''-')(5TT-j)]-:j?i(>-')- 

KXAMPI^KS. 

In  a  manner  similar  to  that  just  pursued  find  the  limits 
in  each  of  the  following  examples : 

-  -';  [(^-,4i)(^-^)] 


302  UNIVERSITY   ALGEBRA 

435.  Sometimes  we  have  to  deal  with  the  quotient  of 
two  expressions,  each  of  which  is  infinite:  Such  an  ex- 

pression  for  example,  as  f-^r j-^-f )   when  x 

approaches  zero.  Here  both  dividend  and  divisor  in- 
creases without  limit  as  x  approaches  zero.  When;r=0 
this  expression  is  void  of  meaning  until  a  meaning  is 
assigned  to  it.  By  the  value  of  this  expression  when 
x=0  we  mean  the  limit  which  this  expression  approaches 
as  X  approaches  zero.  To  find  this  limit  is  the  problem 
before  us. 

436.  Let  us  now  find  the  limit  of  (-^^^-:^(~^\ 

as  X  approaches  zero.     By  successive  reductions  we  have 

/_1 1\  ,  rx''+^\x-(x-l)        X 

U— 1     x)     \     X    )        xlx—l)    x^+S 

1__        X     ^  1 

'^x(x-'l)  x^  +  S^Cx-l^^x-'  +  S)      • 
Therefore  by  theorem  I, 

limit  rf  J__  1  Wf^!±?^l=limit 1 

But  plainly  _J™^q  (x- 1X^2  + 3)  =  -3 

Hence  H-['fe-i)-r-^)]=-J- 

Exampi.es. 

In  a  manner  similar  to  that  just  pursued  find  the  limits 
in  each  of  the  following  examples : 

3      .  o\     fx+1 


THEORY   OF    LIMITS.  303 

■x+1        2    \  12 


,  limit  rf£±i_^_w-^i-l 
^  limit  rf.J__ lUf-ll-^l 

*•  ^  5  4  LV;tr-4    x)    \x^  -WJ 


-.  limit  J  1      1-5- 


CHAPTER  XIX. 
RATIO,  PROPORTION  AND  VARIATION. 

437.  The  relative  magnitude  of  two  numbers  or  quan- 
tities, measured  by  the  number  of  times  the  first  contains 
the  second,  is  called  the  Ratio  of  the  two  numbers  or 
quantities. 

Thus  12  contains  3  just  four  times;  hence  the  ratio  of 
12  to  3  is  4,  or  ^.     And  similarly  if  a  and  b  stand  for 

a7iy  two  numbers  the  ratio  of  ^  to  ^  is  -7. 

0 

438.  We  may  speak  of  the  ratio  of  two  quantities  of 
the  same  kind  as  well  as  the  ratio  of  two  numbers.  Thus, 
12  feet  contains  3  feet  just  4  times,  hence  the  ratio  of  12 
feet  to  3  feet  is  4  or  ^-^-. 

A  good  opportunity  is  here  afforded  to  call  attention  to  the  double 
use  of  the  word  division.  When  a  stick  12  feet  long  is  cut  into  four 
equal  pieces  each  piece  is  a  stick  3  feet  long.  In  this  case  we  say  that 
12  feet  is  divided  by  4.  This  process  is  called  division  —  the  division 
of  separation.  A  stick  3  feet  long  may  be  used  as  a  measure  with 
which  to  measure  a  stick  12  feet  long,  and  the  latter  contains  the 
former  just  four  times.  In  this  case  we  may  say  that  12  feet  is  meas- 
ured by  3  feet.  This  process  is  called  ?neasurement ;  it  is  also  called 
division — the  division  of  measurement.  In  the  division  of  separation 
the  divisor  is  always  a  number,  and  the  quotient  a  quantity  of  the  same 
kind  as  the  dividend.  In  the  division  of  measurement  the  measure  is 
always  'a  quantity  of  the  same  kind  as  the  one  measured  and  the  ratio 
is  always  a  number. 

The  ratio  of  any  two  quantites  of  the  same  kind  may  be  looked 
upon  as  the  numbgr  of  units  in  the  first  divided  by  the  number  of  units 
in  the  second.  Plainly,  quantities  which  are  not  of  the  same  kind 
cannot  have  any  ratio,  for  one  cannot  possibly  be  measured  by  the 
other,  nor  can  one  be  contained  in  the  other.  For  example,  ten  miles 
cannot  be  measured  by  two  quarts,  nor  can  two  quarts  be  contained 
any  number  of  times  in  ten  miles. 


RATIO,   PROPORTION   AND   VARIATION.  305 

439.  The  ratio  of  a  to  d  is  denoted  in  either  of  two 
ways:  yirs^,  by  writing  the  a  before  the  d  with  a  colon 
between  them,  thus,  a:d;  second,  by  a  fraction  in  which 

a  is  the  numerator  and  d  is  the  denominator,  thus,  7- 

a 

Whichever  way  the  ratio  is  written,  it  is  read  '  'the  ratio 
oi  a  to  d,'^  or  simply  ''a  to  d.^^ 

440.  In  either  way  of  writing  the  ratio  of  a  to  d,  a  is 
called  the  Antecedent  or  First  Term,  and  d  is  called 
the  Consequent  or  Second  Term. 

PROPBRTIKS  OP  RATIOS. 

441.  Since  a  ratio  is  the  quotient  obtained  by  dividing 
the  number  of  units  in  the  antecedent  by  the  number  of 
units  in  the  consequent,  it  follows  that  the  properties  of 
ratios  may  be  obtained  immediately  from  the  properties 
of  fractions. 

442.  Since  a  fraction  may  be  multiplied  either  by 
multiplying  the  numerator  or  dividing  the  denominator, 
it  follows  that  a  ratio  may  be  multiplied  either  by  multi" 
plyiyig  the  antecedent  or  by  dividing  the  consequent, 

443.  Since  a  fraction  may  be  divided  either  by  divid- 
ing the  numerator  or  multiplying  the  denominator,  it 
follows  that  a  ratio  may  be  divided  either  by  dividing  the 
antecedent  or  by  multiplying  the  consequent, 

444.  Since  a  fraction  remains  unchanged  in  value 
when  both  numerator  and  denominator  are  multiplied  or 
divided  by  the  same  number,  it  follows  that  a  ratio  remains 
unchanged  in  value  when  both  antecedent  and  consequent  are 
multiplied  or  divided  by  the  same  number, 

20 -U.  A. 


306  UNIVERSITY   ALGEBRA. 

445.  If  the  numerator  of  a  fraction  is  greater  than  the 
denominator,  the  fraction  is  greater  than  1  ;  therefore,  if 
the  antecedent  of  a  ratio  is  greater  than  the  consequent y  the 
ratio  is  greater  than  1.  , 

446.  If  the  numerator  of  a  fraction  is  less  than  the 
denominator,  the  fraction  is  less  than  1 ;  therefore,  if  the 
antecedent  of  a  ratio  is  less  than  the  co7isequenty  the  ratio  is 
less  than  1. 

447.  If  the  numerator  and  denominator  of  a  fraction 
are  equal  to  each  other,  the  fraction  is  equal  to  1 ;  there- 
fore, if  the  antecedent  and  consequent  of  a  ratio  are  equal  to 
each  other y  the  ratio  is  equal  to  1, 

448.  Theorem.  A  ratio  which  is  greater  than  1  is  de- 
creased by  increasing  both  antecedent  and  consequent  by  the 
same  amount. 

Let  7  be  a  ratio  which  is  greater  than  1 ;  then  a^b, 

0 

Now  form  a  new  ratio  by  increasing  the  antecedent  and 
consequent  by  the  same  amount,  x.     The  new  ratio  is 

a-^-x 

b+x 
If  we  multiply  antecedent  and  consequent  of  the  original 

ratio  by  ^+^,  we  get 

a^^ab-j-ax 

b^b^  +  bx'  ^^^ 

If  we  multiply  antecedent  and  consequent  of  the  new 

ratio  by  b,  we  get 

a+x^ab+bx 

b-^-x^b^+bx  ^^ 

Now,    as   «>^,    it  is   plain   that  ax^bx^    and    hence 

ab-j-ax^ab+bx.     Therefore, 

ab-\-ax     ab-\-bx 

b'''^bx^b^-\-bx 


RATIO,   PROPORTION   AND   VARIATION.  307 

Therefore,  from  (1)  and  (2), 

a     a-^x 

which  is  what  was  to  be  proved. 

Since  a>dy  it  is  plain  that  a+x>d+x.     Therefore, 

b+x^  ' 

Therefore,  as  each  ratio  7  and  -j-. —  is  greater  than  1,  and 
o  o-f-x 

as  7>T-^ — '  it  follows  that  7-; —  zs  nearer  the  value  1 
0     b-\-x  o+x 

a 
than  7  is. 
0 

449.  Theorem.  A  ratio  which  is  less  than  1  is  in- 
creased by  increasing  both  antecedent  and  consequent  by  ike 
same  amount, 

a 
Let  7  be  a  ratio  which  is  less  than  1 ;  then  a'>b.   Now 
0 

form  a  new  ratio  by  increasing  the  antecedent  and  conse- 
quent by  the  same  amount,  x.     The  new  ratio  is 

a-\-x 
b+x 
If  we  multiply  antecedent  and  consequent  of  the  original 
ratio  by  ^+;r,  we  get 

a_ab'^ax 

b^b'^+bx  W 

If  we  multiply  antecedent  and  consequent  of  the  new 
ratio  by  b,  we  yet 

a-\-x^ab-j-  bx 

b+x'^b'^+bx  ^"^^ 

Now,  as  a<ib,  it  is  plain  that  ax<.bXy  and  hence 
ab-{-ax<Zab-\-bx.     Therefore, 

ab-\-ax     ab-j-bx 

¥+bi^W+hc' 


308  UNIVERSITY   ALGEBRA. 

Therefore,  from  (1)  and  (2), 

a     a+x 

b^J+'x 
wliicli  is  what  was  to  be  proved. 

Since  a<,by  it  is  plain  that  a+x<^b+x.     Therefore, 

b^x^  ' 

Therefore,  as  each  ratio  -7  and  7- —  is  less  than  1,  and  as 
b  b-\-x 

T<i— — >  it  follows  that  -7— —  is  nearer  the  value  1  than  -  is. 
0     b+x  b-\-x  b 

This  last  statement,  together  with  the  last  statement 
in  the  previous  article,  shows  that  any  ratio  (except  the 
ratio  1)  is  made  more  nearly  the  value  1  by  increasing  both 
antecedent  and  consequent  by  the  same  amount. 

INCOMMENSURABLE  NUMBERS. 

450.  Two  numbers  are  Commensurable  wi^h  each 
other  when  there  exists  some  third  number  which  is  con- 
tained an  INTEGRAI,  number  of  times  in  each  of  the  two 
numbers.  For  example,  14  and  6  are  commensurable 
with  each  other,  because  there  is  a  number,  viz.:  2, 
which  is  contained  7  times  in  14  and  3  times  in  6. 
Again,  f  and  f  are.  commensurable  with  each  other,  for 
there  is  a  third  number,  viz. :  -^,  which  is  contained  14 
times  in  f  and  15  times  in  f . 

The  idea  may  be  expressed  otherwise  by  saying  that 
two  numbers  are  commensurable  with  each  other  when  their 
ratio  equals  the  ratio  oj  two  whole  numbers.  For  example, 
|:f=14:15. 

451.  Two  numbers  are  Incommensurable  with  each 
other  when  there  is  no  number  which  is  contaified  in  each 


Wk 


RATIO,   PROPORTION   AND   VARIATION.  309 

of  the  given  numbers  an  inTKGRAI,  number  of  times ^  or 
what  is  the  same  thing,  two  numbers  are  incommensurable 
with  each  other  when  their  ratio  cannot  be  expressed  as  the 
ratio  of  two  WHOI^K  numbers, 

452.  A  Commensurable  Number  is  one  which  is 
commensurable  with  unity;  and  an 

Incommensurable  Number  is  one  which  is  incom- 
mensurable with  unity. 

Thus,  1/2,  1/5,  1/6  are  examples  of  incommensurable 
numbers. 

453.  From  the  definition  of  an  incommensurable  num- 
ber it  follows  that  the  ratio  of  an  incommensurable 
number  to  unity  cannot  be  expressed  as  a  fraction  with 
an  integral  numerator  and  denominator.  But  the  ratio 
of  any  number  to  unity  is  that  number  itself.  Therefore, 
an  incommensurable  number  cannot  be  expressed  by  a 
fraction  with  an  integral  numerator  and  denominator. 
Since  any  decimal  which  terminates  can  be  expressed  as 
a  fraction  with  a  whole  number  for  numerator  and  denom- 
inator, therefore  an  incommensurable  number  cannot  be 
expressed  as  a  decimal  which  terminates, 

454.  But  while  we  cannot  find  either  a  decimal  or  a 
common  fraction  with  an  integral  numerator  and  denom- 
inator which  will  exactly  express  any  given  incommen- 
surable number,  still  we  can  find  either  a  decimal  or  a 
common  fraction  which  will  differ  from  the  given  incom- 
mensurable number  by  an  amount  as  small  as  we  please. 

Thus,  1/2  is  an  incommensurable  number,  but,  if  we 
wish,  we  may  write  a  number  which  will  differ  from  V  ^ 
by  less  than  one  ten  thousandth  ;  the  number  is  1.4142. 
We  may  write  a  number  which  will  differ  from  l/2  by 


3IO  UNIVERSITY    ALGEBRA. 

less  than  one  millionth;  the  number  is  1.414213.  A  still 
closer  approximation  to  l/2  is  1.41421356,  but  this  is 
not  exactly  equal  to  1/2. 

455.  It  has  been  stated  that  an  incommensurable 
number  cannot  be  expressed  as  a  decimal  which  termi- 
nates. But  the  value  of  an  incommensurable  number  may 
be  expressed  decimally  to  a  greater  and  greater  degree 
ot  accuracy  by  carrying  it  out  to  a  greater  and  greater 
number  of  decimal  places.  We  may  then  say  that  an 
incommensurable  num^ber  is  a  never  ending  decim>al. 

456.  Repeating  decimal  represents  a  Commen- 
surable Number.  Consider,  for  example,  the  repeating 
decimal  .126126,  ....*,  and  let  the  value  of  this  be 
represented  by  a.     Hence, 

a=. 126126 

Multiplying  this  equation  by  1000,  we  have 

1000^=126.126126  .... 
Subtracting  126  from  each  member,  we  have 

1000a- 126=.  126126  .... 
Therefore,  1000a-126=«. 

Hence,  999^=126. 

Hence,  «=Tll=rnr- 

In  a  similar  manner  it  may  be  shown  that  any  repeat- 
ing decimal  is  expressible  as  a  common  fraction  with  an 
integral  numerator  and  denominator.  A  repeating  decimal 
therefore  cannot  be  an  incommensurable  number.  Whence 
we  conclude  that  an  incommensurable  number  is  a  never 
ending  decimal  which  does  not  repeat. 

Some  incommensurable  numbers  have  been  computed 
to  a  great  many  decimal  places.     This  is  especially  the 

♦When  dots  of  continuation  are  used  in  this  way  it  is  understood  that  the 
number  represented  is  the  exact  number  of  which  the  portion  ah  eady  written 
is  only  an  approximation,  and  that  closer  and  closer  approximations  are  given 
by  taking  more  and  more  figures  in  the  decimal. 


RATIO,   PROPORTION    AND  VARIATION 


311 


case  with  the  incommensurable  number  which  represents 
the  circumference  of  a  circle  whose  diameter  is  unity. 
This  number,  carried  out  to  thirty  decimal  places,  is  as 
follows :   3.14159265358979328462643383280. 


457.  It  has  been  shown  that  an  incommensurable 
number  cannot  be  expressed  as  a  fraction  with  an  integral 
numerator  and  denominator,  that  it  cannot  be  expressed 
as  a  decimal  'which  terminates,  and  that  it  cannot  be 
expressed  by  a  repeating  decimal.  The  student  may  infer 
from  all  this  that  an  incommensurable  number  is  not  an 
exact  number  at  all,  but  such  is  not  the  case,  as  may 
easily  be  shown  in  the  case  of  the  incommensurable  num- 
ber l/2,  for  we  can  draw  a  geometric  representation  of  l/2. 
Take  each  of  the  two  sides,  CA  and  CB,  of  a  right  angled 
triangle  equal  to  1.     Then  AB,   the  hypotenuse,    will 

equal  V {ly  +  iXy^V^.  Thus  1^2  is  the  exact  distance 

from  A  to  B,  which  is  a  perfectly  A^ 

definite  distance.     Thus  the  idea 

that  incommensurable   numbers 

are  indefinite  or  inexact  must  be 

avoided.    This  notion  has  arisen  (j) 

because  \h^  fractions  ^e  often  use 

in    place     of    incommensurable 

numbers,  such   as  1.4142+    for 

l/2,    are   merely  approximations    Q  (I)  iB 

to  the  true  values. 

COMPOUND  RATIOS. 

a  c 

458.  If  from  two  given  ratios,  ~  and  -^  or  a\b  and  c\  dy 

we  form  another  ratio  by  multiplying  the  antecedents 
together  for  a  new  antecedent  and  the  consequents  to- 


312  UNIVERSITY   ALGEBRA. 

CiC 

getter  for  a  new  consequent,  we  get  -j-  or  ac\bd,  which 
is  said  to  be  Compounded  of  the  given  ratios  a :  b 
and  c :  d, 

459.  If  ^>1  it  follows  that  xy^y,  or,  expressed  in 
words,  if  the  multiplyer  is  greater  than  1  the  product  is 
greater  than  the  multiplicand. 

Therefore,  if  ^>l,^J>f  Also,  if  J>1.  |"^>i. 

n    ^  a  c       ac ,      ^ 

But  in  each  of  these  cases  t  3  o^  — /  is  the  ratio  com- 

0  a       ca 

a       ^  c 
pounded  of  the  ratios  7  and  — 

Therefore,  if  a  ratio  be  compounded  of  two  ratios  each  of 
which  is  greater  than  1  the  result  is  greater  than  eithef  of 
the  given  ratios, 

460.  In  a  similar  way  it  may  be  shown  that  if  a  ratio 
be  compounded  of  two  ratios  each  of  which  is  less  than  1  the 
result  is  less  than  either  of  the  given  ratios, 

461.  If  x<X  it  follows  that  xy<^y,  or,  expressed  in 
words,  if  the  multiplyer  is  less  than  1  the  product  is  less 
than  the  multiplicand. 

Therefore,  if  ^<1)  Ijl^T  ^^^  ^^  ^^^  before  shown 
,n  ^     H    etc ^  c 

Therefore,  if  a  ratio  be  compounded  of  two  given  ratios 
one  of  which  is  greater  than  1  and  the  other  less  than  1,  the 
7'esult  is  intermediate  in  value  between  the  two  given  ratios. 


RATIO,   PROPORTION    AND   VARIATION.  313 

PROBLKMS. 

1.  Arrange  the  ratios  2  : 3,  3  :5,  5  :8  in  the  order  of 
magnitude. 

2.  Which  is  nearer  unity  2  : 3  or  8 : 9? 

3.  Which  is  nearer  unity  2  :3  or  2+;tr:3+^? 

4.  For  what  value  of  x  will  the  ratio  8+jr:36+;r  be 
equal  to  the  ratio  1:3? 

5.  What  must  be  added  to  each  term  of  the  ratio  9 :  16 
to  produce  the  ratio  3:4? 

6.  What  must  be  subtracted  from  each  term  of  the 
ratio  3  : 5  to  produce  the  ratio  5:9? 

7.  A  certain  ratio  will  become  equal  to  \  when  1  is 
subtracted  from  each  of  its  terms,  and  equal  to  f  when  9 
is  added  to  each  of  its  terms.     Find  the  ratio. 

8.  Find  two  numbers  such  that  if  each  is  increased  by 
1  the  results  have  the  ratio  2 : 3,  and  if  each  is  increased 
by  7  the  results  have  the  ratio  5:6. 

9.  Find  two  numbers  such  that  if  the  first  is  increased 
by  2  and  the  second  decreased  by  2  the  results  have  the 
ratio  6:5  but  if  the  first  is  decreased  by  2  and  the 
second  increased  by  2  the  results  have  the  ratio  4 : 7. 

10.  A  rectangle  is  39  feet  long  and  36  inches  wide. 
Express  the  ratio  of  the  length  to  the  breadth. 

11.  What  is  the  ratio  of  12  lbs.  8  oz.  to  21  lbs.  14  oz.? 

12.  What  is  the  ratio  of  5  ft.  to  6  ft.  3  in.? 

13.  Two  rectangular  fields  have  the  same  area.  The 
length  of  the  first  is  180  feet  and  of  the  second  150  feet, 
what  is  the  ratio  of  their  widths? 

14.  The  areas  of  two  rectangular  rooms  have  the  ratio 
2:3.  The  length  of  the  first  is  12  feet,  and  of  the  second 
20  feet.     What  is  the  ratio  of  their  widths? 


314  UNIVERSITY    ALGEBRA. 

15.  Two  numbers  are  in  the  ratio  of  3:4,  and  the  sum 
of  their  squares  is  400.     What  are  the  numbers? 

16.  Two  equal  sums  of  money  are  on  interest;  the 
first  runs  8  months  at  7  per  cent.,  the  second  runs  9 
months  at  6  per  cent.  The  interest  in  the  first  case  has 
what  ratio  to  the  interest  in  the  second  case? 

17.  Divide  the  number  10  into  two  such  parts  that  the 
squares  of  these  parts  shall  have  the  ratio  9 : 4. 

18.  A  train  of  cars  travels  140  miles  in  3^  hours,  and 
another  train  travels  240  miles  in  5  hours.  What  is  the 
ratio  of  their  rates  of  speed? 

19.  Show  that  the  ratio  aid  is  equal  to(a+xy:(d+ x)  ^ 
iix'^=^ab, 

20.  What  must  be  the  value  of  x  in  order  that  the 
ratio  &\x  may  equal  b—x} 

PROPORTION. 

462.  The  expression  of  equality  which  exists  between 
two  ratios  is  called  a  Proportion.  Thus,  a  : b=c  \d  is  a 
proportion.  In  this  case  the  four  numbers  or  quantities 
are  said  to  be  in  proportion. 

463.  Sometimes  the  proportion   is  expressed  as  an 

ordinary  equation  in   fractions,  thus,   7-= "3'  ^^^  some- 

o      a 

times  by  the  notation  a\b\\c\d.     However  written,  the 
proportion  is  read,  ''a  is  to  ^  as  ^  is  to  d, 

464.  In  the  proportion  a  :  b=c\dWMt  letters  a,  b,  c  and 
d  are  called  the  Terms  of  the  proportion.  The  first  and 
fourth  terms  are  called  the  Extremes,  and  the  second 
and  third  terms  are  called  the  Means. 


RATIO,   PROPORTION   AND   VARIATION.  315 

465.  If  «,  b,  c,  and  d  are  proportional,  we  have  the 
proportion  a\b\\c\d,ox,  written  in  the  fractional  form. 

Of       c 

-=v     Now,  multiplying  each  member  by  hd,  we  get 
0     a 

ad=  be. 
That  is,  the  product  of  the  means  equals  the  product  of  the 
extremes, 

466.  If  we  have  given  the  equation 

ad^=bc, 
then  dividing  each  member  by  bd,  we  get 
ad    be      a      c  '     . 

Hence,  tf  the  product  of  two  numbers  equals  the  product  of 
two  other  numbers,  the  numbers  are  in  proportion^  or,  in 
other  words,  if  the  product  of  the  two  numbers  equals  the 
product  of  two  other  numbers,  the  factors  of  one  product  may 
be  taken  for  the  extremes  and  the  factors  of  the  other  product 
may  be  taken  for  the  means  of  a  proportion, 

467.  From  the  equation  ad=be  we  may  either  infer 

a\b=c:d  (1) 

or  a:e=^b:d  (2) 

or  b:a=d:c  (3) 

or  b:d=a',  c  (4) 

and  therefore  from  (1)  we  mayinfer  either  (2)or  (3)  or  (4). 
The  proportion  (2)  is  said  to  be  deduced  from  (1)  by 
Alternation.  The  proportion  (3)  is  deduced  from  (1) 
by  Inversion.  The  proportion  (4)  is  really  the  same  as 
(2)  except  that  the  two  members  have  changed  places ; 
it  may  be  deduced  from  (3)  by  alternation. 


3l6  UNIVERSITY   ALGEBRA. 

468.  If  a\b=c:dy  then  writing  in  fractional  form, 

a c 

IT'd 
Adding  1  to  each  member, 

or  in  another  form,        — r-  =  — r"' 

0  a 

or  in  the  common  form  of  proportion, 
a-\-b\b=c-\-d\d. 
This   last    proportion,    a-\'b\b=^c-\-d\d,  is  said  to  be 
derived  from  the  proportion  a:b—c:dhy  Composition. 

a     c 

469.  If  a:b=c:dj  then  j^  =  y  and  subtracting  1  from 

each  member,  7  — 1=-— 1, 

0  a 

a — b    c—d 
or  m  another  form,        — ^= — r-> 
0  a 

or  in  the  common  form  of  proportion, 

a — b'.b=c — d:d. 

This  last  proportion,  a—b:b=c—d:d,    is   said   to  be 

derived  from  the  proportion  a  :b=c:d  by  Division. 

470.  If  a'.b=c:d,  then  by  composition, 

a-\-b     c+d 

and  by  division,  ~~b~^~d~'  ^^-^ 

Divide  (1)  by  (2)  member  by  member  and  we  get 
a-\-b^c+d 
a — b    c — d 
or  written  in  the  ordinary  form  of  proportion, 
a-\-b\a—b^c-\rd\  c—d. 
This  last  proportion,  a-\-b:a — b=c+d:c—d,  is  said  to 
be  derived  from  the  proportion  a:b=c',dhy  Composition 
and  Division. 


RATIO,   PROPORTION    AND    VARIATION.  317 

471    The  products  of  corresponding  terms  of  two  or  more 
sets  of  proportional  nuTubers  are  proportional. 

Let  a  :  d=c  :  d 

and  e  :f=k  \k 

and  n  :  r=  s  :  t 

Writing  each  of  tliese  proportions  as  an  equation  in 
fractions,  we  have 

a__c 

ir~d. 

e^h 

n__s 
r~^~t 

and  from  these  equations  by  multiplication  we  obtain 

a e n  ^c h s 

'bf~r~dkl 
aen__chs 
^^  'bfr'^m 

or  writing  this  in  the  ordinary  notation  of  proportion, 
we  have  aen  :  dfr=cks  :  dkty 

which  is  what  was  to  be  proved. 

472.   I^ike  powers  or  like,  roots  of  proportional  numbers 
are  proportional, 

Let  a  :  b=c  :  d. 

Writing  this  proportion  as  an  equation  in  fractions,  we 

a      c 
obtam  I^T  ^^ 

and  raising  each  member  to  the  n  th  power,  we  have 

a""     <f' 

jn-=jn^  (2) 

or  writing  this  in  the  ordinary  notation  of  proportion, 
we  have  a'^^«=^":^".  (3) 


31 8  UNIVERSITY   ALGEBRA. 

Again,  taking  the  n  th  root  of  each  member  of  (1),  we 

1       1 

a*^  ctt 
have  -1=-,  (4) 

bn  dn 
or  writing  in  the  ordinary  notation  of  proportion,  we  have 

1      1        L       i 

an\  b't=^c*^\  ^«.  (5) 

Equations  (3)  and  (5)  are  those  which  were  to  be  proved. 

473.  In  the  proportion  a  :  ^=^  :  ^,  ^  is  called  a  Mean 
Proportional  between  a  and  c,  and  c  is  called  a  Third 
Proportional  to  a  and  b. 

If  in  this  proportion  we  write  the  product  of  the  means 
equal  to  the  product  of  the  extremes,  we  get 

b'^^ab, 
or  extracting  the  square  root  of  each  member,  we  get 

Therefore,  a  mean  proportional  between  two  members  is 
equal  to  the  square  root  of  their  product. 

474.  A  Continued  Proportion  is  a  series  of  equal 
ratios.     Thus,         a  :  b=c  :  d=^e  :/,  etc. 

475.  In  any  continued  proportion  any  antecedent  is  to 
its  corresponding  consequent  as  th^  sum  of  all  the  antecedents 
is  to  the  sum  of  all  the  consequents. 

Let  the  continued  proportion  be  written  as  several 
equal  fractions,  thus : 

a __c ^  e 

and  let  r  be  the  common  value  of  each  of  these  fractions, 
then 

—=r  or  a=br, 
o 

— ==r  or  c=dr, 
d 

-.  =  r  or  e^=fr. 


RATIO,   PROPORTION    AND    VARIATION.  319 

Therefore,  by  addition, 

aJrc-\-e-=^br-\-dr-\-fr=^{b-\-d^-fy. 
Hence,  by  dividing  by  b-\-d-\-f, 

a-\-c-\-e__  __a 

which  is  what  was  to  be  proved. 

EXAMPI.KS   AND   PROBI^EMS. 

1 .  Write  two  proportions  from  the  equation  5x8=4x10 

2.  Write  two  proportions  from  the  equation  ab^cd. 

3.  Write  two  proportions  from  the  equation  ab=^ac. 

4.  Write  two  proportions  from  the  equation  x'^=yz. 

5 .  Write  two  proportions  from  the  equation  (a  -f  ^)  ^  =  nr, 

6.  Write  two  proportions  from  the  equation  ^n  ^  r=  Apq. 

7.  Form  a  proportion  with  the  numbers  3,  5,  21,  35. 

8.  Form  a  proportion  with  the  numbers  3,  20,  90,  600. 

9.  Form  a  proportion  with  the  numbers  3,  6,  20,  40. 
ID.  What  must  x  stand  for  in  order  that  jt  :  3=6  :  9  may 

be  a  true  proportion  ? 

11.  What  is  the  value  of  ;ir  if  5  :  x==^  :  12  ? 

12.  What  is  the  value  of  ^  if  12  :  15=;*; :  35  ? 

13.  What  is  the  value  of  :r  if  8  :  10=90  :  x  ? 

14.  What  is  the  value  of  ;t:  if  ;r  :  6=;i;— 1  i^f  +  ll? 

15.  What  is  the  value  of  ;r  if 

lb(a-\-b)     10{aby^a-b  ^     ^ 
14.{a-b)  '21(aby     a+b'^' 

16.  What  is  the  value  of  ;ir  if 

17.  What  is  the  value  of  x  if 

x:  (5;^-6r)  =  (3;^2^2;/r— 8r2)  :  (5;^2^4«r~12r2)? 

18.  What  is  the  value  of  x  if 


320  UNIVERSITY   ALGEBRA. 

19.  If  a  :  b=-c :  d,  prove  a  :  mb=c :  md, 

20.  li  a:  b=c :  d,  prove  na  :  rb=nc :  r^. 

21.  If  a  :  ^=^:^,  prove 

{na-\-rb')  :  (na — rb)  =  (nc-\-rd)  :  (nc—rd). 

22.  If  a  :  <?=^ :  d,  prove  ;2«  :  r^=-  :  -• 

r    n 

23.  If  ^  :  b=c:  ^, prove  that  a  :a+c=a  +  b  \a  +  b+c-\-d. 

24.  If  <2  :  b'=c :  d,  prove  that 

a'^-i-ab+b'^  :  a'^—ab+b^=-c^+cd+d^  :c^^cd+dK 

25.  Prove  that  a  :  b=c  :  d,  if 

(^  +  ^4-^+^)  (^_-<^— ^-f  ^)  =  (^a—b+c—d)  (a  +  b—c—d). 

VARIATION. 

476.  One  number  or  quantity  is  said  to  Vary  Directly 
as  another  when  the  number  of  units  in  the  first  is  equal 
to  the  number  of  units  in  the  second  multiplied  or 
divided  by  some  constant  number,  or  what  is  the  same 
thing,  when  the  ratio  of  the  number  of  units  in  the  first 
to  the  number  of  units  in  the  second  is  some  constant 
number. 

For  example,  if  a  train  of  cars  travels  at  the  rate  of 
fort}^  miles  an  hour,  the  number  of  miles  traveled  varies 
directly  as  the  number  of  hours  occupied.  Now  we  may 
take  a  letter,  say  x,  to  represent  the  number  of  miles 
the  train  travels,  and  another  letter,  say  j/,  to  repre- 
sent the  number  of  hours  the  train  travels  ;  then,  plainly, 

we  have  x==40y  or  —=40. 

477.  The  word  directly  is  often  omitted,  and  we  say 
simpl}^  that  one  number  or  quantity  varies  as  another. 
The  same  idea  is  often  expressed  by  saying  that  the 
second  number  or  quantity  is  Proportional  to  the  first 
number  or  quantity.     In  the  above  illustration  the  dis- 


RATIO,    PROPORTION    AND    VARIATION.  32 1 

tance  traveled  is  proportional  to  the  time  occupied,  or  the 
number  of  miles  traveled  is  proportio7ial  to  the  number  of 
hours  occupied. 

X 

478.  If  X  is  proportional  to  y  then  -  =^  where  c  is  some 

constant,  i.  e.  some  number  that  remains  unchanged. 
Now,  if  we  represent  some  particular  value  of  x  by  jt^, 
and  the  corresponding  particular  value  oi  y  byjFi,  then 

X 

-^  —  c  and  from  this  it  is  easily  seen  that  when  one  num- 

ber  is  proportional  to  another,  we  know  the  constant  by 
which  one  of  the  numbers  must  be  multiplied  to  produce 
the  other,  if  we  know  any  corresponding  particular  values 
of  the  two  numbers.  For  example,  if  we  know  that  a 
traveler  is  walking  uniforml}^,  i.  e.  at  the  same  rate,  then 
we  know  that  the  number  of  miles  traveled  is  propor- 
tional to  the  number  of  hours  occupied.  Hence  if  we 
represent  the  number  of  miles  traveled  by  x  and  the 
number  of  hours  occupied  by  y,  we  have 

X 

x=cy  or  -=c. 

y 

Now  if  we  further  know  that  the  traveler  walks  15  miles 

in  5  hours,  we  have  r=-^  =  3. 

From  this  we  see  that  3  may  be  written  in  place  of  Cy  and 

X      ^ 
hence  we  know  :r=3y  or  -  =3. 

y 

479.  One  number  or  quantity  is  said  to  Vary  In- 
versely as  another  when  the  first  is  equal  to  some  con- 
stant divided  by  the  second.     Thus,  x  varies  inversely 

as  r,  if  x^=—  where  c  is  some  constant. 

y 

480.  If  in  the  equation  ;r=-we  multiply  each  number 
by  j^,  we  get  xy^=^c. 


322  UNIVERSITY   ALGEBRA. 

Therefore,  we  say  that  one  number  varies  inversely  as  a 
second  when  the  product  of  the  two  numbers  is  a  constant. 
For  example,  the  time  occupied  in  traveling  a  certain 
distance  varies  inversely  as  the  rate  of  speed,  or  the  number 
of  hours  occupied  in  traveling  a  certain  number  of  miles 
varies  inversely  as  the  number  of  miles  traveled  per  hour. 

481.  Instead  of  saying  that  one  number  varies  inversely 
as  another,  the  same  idea  is  often  expressed  by  saying  that 
one  number  is  Reciprocally  Proportional  to  another. 

482.  One  number  Varies  Jointly  as  two  others  when 
the  first  is  equal   to   some  constant   multiplied   by  the 

product  of  the  other  two.     Thus,  if  x=^cyz,  or  — =^, 

yz 

where  c  is  some  constant,  x  is  said  to  vary  jointly  as 
y  and  z.  The  same  idea  is  often  expressed  by  saying  that 
one  number  is  Proportional  to  the  Product  of  two  others. 

483.  One  number  is  said  to  vary  as  the  square  of 
another  when  the  first  is  some  constant  multiplied  by  the 
square  of  the  second,  or  when  the  first  divided  by  the 
square  of  the  second  is  some  constant.     Thus,  if  x^^cy'^, 

X 

or  -2=^,  where  c  is  some  constant,  x  is  said  to  vary  as  jk^. 

The  same  idea  is  often  expressed  by  saying  that  one 
number  is  Proportional  to  the  Square  of  another. 

484.  One  number  is  said  to  Vary  Directly  as  a  second 
and  Inversely  as  a  third  when  the  first  is  equal  to  some 
constant  multiplied  by  the  ratio  of  the  second  to  the  third. 

Thus,  if  x=^c—^  X  is  said  to  vary  directly  as  y  and  in- 
versely as  z. 

The  same  idea  is  often  expressed  by  saying  that  the 
first  number  is  Directly  Proportional  to  the  second 
and  Inversely  Proportional  to  the  third. 


RATIO,   PROPORTION    AND    VARIATION.  323 

KXAMPLES   AND   PROBLEMS. 

1.  If  ;ir  is  proportional  to  jk,  and  x=Ab  when  j|/=3,  find 
the  value  oi  x  when_y=15. 

2.  If  ;r  is  inversely  proportional  to  y,  and  ^=1  when 
y=^,  find  the  value  of  x  when  y=lb. 

3.  Form  a  proportion  with  the  two  sets  of  values  of  x 
and  y  in  problem  2. 

4.  If  -T  varies  as  jj/*'^,  and  ;i:=l  when  y—h,  find  the 
value  of  X  when  ^=  15. 

5.  Form  a  proportion  with  the  two  sets  of  values  of 
X  and  jj/^  in  problem  4. 

6.  If  j^;  varies  jointly  as  y  and  2,  and  ;i;=20  when  j/=2 
and  >2'=3,  find  the  value  of  ;r  whenjj/=3  and  -3'=  6. 

7.  If  ;t:  varies  inversely  as  the  square  oi  y,  and  jr=l 
when  jK=10,  find  the  value  of  :r  when  jr=5. 

8.  If  ;t:  is  directly  proportional  to  y  and  inversely  pro- 
portional to  2,  and  ;»;=20  when  y=Q  and  -3*= 4,  find  the 
value  of  ^  when  j/=3  and  -3'=  10. 

g.   li  X  varies  asj^,  prove  that  x^  varies  asjj/^. 

10.  If  X  is  inversely  proportional  to  y  and  j/  is  inversely 
proportional  to  2,  pro\^e  that  x  is  proportional  to  z. 

11.  If  ;»;  is  proportional  to  2  and  j|/  is  also  proportional 
to  2,  prove  that  xy  is  proportional  to  2'^,  also  that  x'^-^y^ 
is  proportional  to  2'^, 

12.  If  Sx+7y  is  proportional  to  3;r+13j/  and  ;r=5 
whenjK=3,  find  the  ratio  oi  x  to  y  and  thus  show  that  x 
varies  as  y. 

13.  The  number  of  feet  a  body  falls  is  proportional  to 
the  square  of  the  number  of  seconds  occupied  in  falling. 
Knowing  that  a  body  falls  16  feet  the  first  second,  find 
liow  many  feet  it  will  fall  in  5  seconds. 


324  UNIVERSITY    ALGEBRA. 

14.  With  the  same  supposition  as  in  the  last  example, 
find  the  height  of  a  tower  from  which  a  stone  dropped 
from  the  summit,  reaches  the  ground  in  3f  seconds. 

15.  The  w^eight  of  a  metal  ball  is  proportional  to  the 
cube  of  the  radius,  and  a  ball  whose  radius  is  2  inches 
weighs  10  pounds,  what  is  the  weight  of  a  ball  whose 
radius  is  5  inches? 

16.  If  a  heavier  weight  draw  up  a  lighter  one  by 
means  of  a  cord  passed  over  a  fixed  wheel,  the  number  of 
feet  passed  over  by  each  weight  in  any  given  time  varies 
directly  as  the  difference  of  the  weights,  and  inversely  as 
the  sum  of  the  weights.  If  10  pounds  draw  up  6  pounds 
16  feet  in  2  seconds,  how  high  will  14  pounds  draw  10 
pounds  in  2  seconds  ? 


CHAPTER   XX 

PROGRESSIONS. 

485.  An  Arithmetical  Progression  is  a  series  of 
terms  such  that  each  term  differs  from  the  immediately 
preceding  term  by  a  fixed  number,  called  the  Common 
Difference.  The  following  are  examples  of  arithmetical 
progressions : 

(1)  2,  4,  6,  8,  10.  (3)  2^,  8f,  5,  6i    7^. 

(2)  31,  26,  21,  16.  (4)  (^-y),  x,  (x+j). 

(5)  a,   (a+d-),   («  +  2^),   Ca  +  Sd), 

486.  The  first  and  last  terms  of  any  given  progression 
are  called  the  Extremes,  and  the  other  terms  are  called 
the  Means. 

487.  The  Arithmetical  Mean  of  two  numbers  a  and 
d  is  found  as  follows:  Let  A  stand  for  required  mean. 
Then,  by  definition     A—a=d—A, 

whence  A=^(a-j-h).  [1] 

Thus,  the  arithmetical  mean  of  7  and  11  is  9,  for  7,  9,  11 
is  an  arithmetical  progression  having  the  common  differ- 
ence of  2. 

By  the  arithmetical  mean  or  average  af  several  7itimbers  is  meant 
something  entirely  disassociated  from  arithmetical  progression.  The 
term  means  the  result  found  by  adding  several  numbers  and  dividing 
by  the  number  of  them.  Thus,  the  arithmetical  mean  of  1.06,  1.21, 
1.93,  is  1.40. 

488.  The  n  th  Term.  It  is  usual  to  represent  the 
first  term  of  an  arithmetical  progression  by  a,  the  common 


326  UNIVERSITY   ALGEBRA. 

difference  by  d,  and  the  n  th  term  by  /.     With  this  nota- 
tion we  may  represent  any  arithmetical  progression  by 
No.  of  term:     1  2  3  4  5  .  .  . 

Progression:  a,  (a-\-d),  {a-{-2d),  (a  +  3^),  (a-j-4d) 
We  notice  that  the  coefficient  of  d  in  the  2d  term  is  1, 
in  the  3d  term  is  2,  in  the  4th  term  is  3,  and,  by  the 
nature  of  the  progression,  the  coefficient  of  d  in  any  term 
is  1  less  than  the  number  of  that  term.  Therefore,  the 
n  th  term  in  this  progression  will  be 

a+(n—l)d, 
or,  representing  the  n  th  term  by  /,  we  have  the  formula 
l=a-\- {n—])d.  [2] 

489.  Evidently  the  sum  of  an  arithmetical  progression 
is  not  changed  if  the  order  of  the  terms  be  reversed;  thus, 

3-f  5+7  +  9-f  11  may  be  written  11+9+7  +  5+3 
2i+3i+4+4f  may  be  written  4f +4+3^+2^ 

in  which  case  the  first  term  becomes  the  last  term,  the 
last  term  becomes  the  first  term,  and  the  common  difference 
cha7iges  signs. 

490.  In  case  the  common  difference  is  positive  the 
progression  may  be  called  an  Increasing  Progression, 
and  in  case  the  common  difference  is  negative  the  pro- 
gression may  be  called  a  Decreasing  Progression. 

491  The  Sum  of  n  Terms.  If  ^  stands  for  the  sum 
of  n  terms  of  an  arithmetical  progression,  we  may  write 
the  two  following  equalities,  the  progressions  being  alike 
except  written  in  reverse  order  : 

^=a+(a+^)  +  (a  +  2^)  +  (a  +  3^)  +  .  .  .+a  +  (in'-l)d  (1) 
^=/+(/_^)  +  (/_2^)  +  (/-3^)  +  .  .  .+l-(n-l)d  (2) 


PROGRESSIONS.  32/ 

Adding  (1)  and  (2)  together  term  for  term,  noticing 
that  the  terms  containing  the  common  difference  nulify 
one  another,  we  have 

2^=(^+/)  +  («4-/)  +  («+/)  +  (^+/)  +  .  .  .  +  (^+/). 
Since  the  number  of  terms  in  the  original  progression 
was  called  n,  we  write  the  last  equation 

whence  the  formula  for  s, 

8=i.n(a.+i),  [3] 

492.  Formula  [2]  enables  us  to  obtain  the  value  of  I 
when  a,  n,  and  d  are  given,  or  the  value  of  a  when  /,  n, 
and  d  are  given,  or  the  value  of  a  when  /,  n,  and  a  are 
given,  or  the  value  of  n  when  /,  a,  and  ^are  given.  Thus: 

(1)  Find  the  20th  term  of  3 +  8+ 13  +  .  .  . 
Here  «=3,  ^=5,  n=20,  therefore 

/rr:3  +  19X5--=98. 

(2)  Find  the  number  of  terms  in  the  progression  5  +  7+9-1-.  .  .+37. 
Here  a==5,  d=2,  1=37,  whence 

37=5  + (;^- 1)2. 
Solving  for  n,  n=Vi. 

(3)  Find  the  common  difference  in  a  progression  of  11  terms  in 
which  the  extremes  are  |  and  30|. 

Here  a=\,  l=30\  and  n=ll,  whence 

301=1 +(11- IK 

Solving  for  cl,  d=3. 

(4)  Insert  3  arithmetical  means  between  5  and  21. 
Here  n=5,  a=5,  and  /=21,  whence 

21=5  +  {5-lK 
Solving  for  ^,  ^=4. 

Therefore,  the  means  are  9,  13,  and  17. 

493.  Formula  [3]  enables  us  to  find  any  one  of  the 
numbers  s,  n,  a  and  /,  when  the  value  of  the  other  three 
are  given.     Thus : 


328  UNIVERSITY   ALGEBRA. 

(1)  Find  the  sum  of  10  terms  of  the  progression  in  which  5  is  the  first 
term  and  -  58  the  last  term. 

Here  «=5,  n=10,  /=  — 58,  whence 

j=iXlO(5-58.) 
That  is,  s=—2Gd. 

(2)  Find  the  number  of  terms  in  an  arithmetical  progression  in 
which  the  first  term  is  4,  the  last  term  22,  and  the  sum  91. 

Here  a=4,  1=22,  and  j=91,  whence 

dl=^n{4  +  22.) 
Solving  for  «,  n='7. 

494.  The  two  formulas 

'  U=a-\-(n-l)d  (I) 

\s=\7i{a  +  l^  (2) 

contain  five  different  letters,  hence  if  any  two  of  them 
stand  for  unknown  numbers,  and  the  values  of  the  rest 
are  given,  the  value  of  the  two  unknown  numbers  can  be 
obtained  by  the  solution  of  a  system  of  two  equations. 
Thus  : 

(1)  Find  the  sum  of  an  arithmetical  progression  in  which  the  last 
term  is  149,  the  common  difference  7,  and  the  number  of  terms  22. 

Here  /=149.  d=l,  and  ?2=22,  whence 

j  149=«  +  (22-l)7  (1) 

\      j=iX22(«  +  149.)  (2) 

From  (1),  a=2. 

Substituting  in  (2)  .r=1661. 

(2)  Find  the  first  term  of  an  arithmetical  progression  of  21  terms, 
whose  sum  is  1197  and  common  difference  is  4. 

Here  «=21,  J=1197  and  d=4:,  whence 

j         /=^+(21-l)4  (1) 

1  1197=iX21{a+/).  (2) 
From  (1),                              l=a  +  m. 

From  (2),  /=114-«.  (3) 

Whence.  /^97  and  ^=17.  (4) 


PROGRESSION.  329 

(3)  Find  the  number  of  terms  in  a  progression  whose  sum  is  1095, 
the  first  term  is  38  and  the  difference  is  5. 
Here  ^=1095,  a=3S  and  d=5,  whence 

j         /=38+(;z-l)5  (1) 

\  1095=J;/(38+/)  (2) 

From  (1),                             /=:334-5;^.  (3) 

From  (2),                       21dO=SSn+nl.  (4) 

Substituting  the  value  of  /  from  (3)  in  (4)  we  get 

2190=71;/+5;/3.  (5) 

Solving  this  quadratic  equation,  we  find 

n=15  or  —29.2. 
The  second  result  is  inadmissable,  since  the  number  of  terms  can 
not  be  either  negative  or  fractional. 

KXAMPI^KS   AND   PROBI.BMS. 

Solve  each  of  the  following  : 

1.  Given  a=7,  ^=4,  7^=13;  find  /and  s. 

2.  Given  a=^,  d=6,  n=SO;  find  /and  s, 

3.  Given  a=9,  /=162,  n==52;  find  s  and  ^. 

4.  Given  a=57,  /=30,  ;z=19;  find  ^and  d. 

5.  Given  /=242,  d=21,  n=12;  find  a  and  s. 

6.  Given  /=16,  d=—S,  n=5S;  find  a  and  s. 

7.  Given  a=17,  /=350,  <3f=9;  find  n  and  s. 

8.  Given  ^=—28,  1=28,  d=7;  find  n  and  s. 

9.  Given  a=l^,  /=54,  ^=999;  find  ^^  and  (sT. 

10.  Given  a=5,  /=16i,  ^=3320;  find  7i  and  ^. 

11.  Given  a=3,  ;z=50,  ^=3825;  find  /  and  ^. 

12.  Given  «=— 45,  n=Sly  s=0;  find  /and  <^. 

13.  Given  /=49,  n=19,  s==50S^;  find  a  and  ^. 

14.  Given  /=105,  72=16,  ^=840;  find  a  and  ^. 

15.  Given  n=S5,  ^=2485,  d=S;  find  <3j  and  /. 

16.  Given  7z=25,  ^=—25,  d=^;  find  ^  and  /. 

17.  Given  ^=4784,  a=41,  ^=2;  find  /  and  72. 


330  UNIVERSITY    ALGEBRA. 

i8.  Given  ^=624,  a=9,  d=4;  find  /and  n, 
ig.  Given  5=278,  d=5,  /=77;  find  a  and  n. 

20.  Given  ^=1008,  ^=4,  /=88;  find  a  andn, 

21.  What  is  the  sum  of  the  first  200  natural  numbers? 

22.  What  is  the  sum  of  the  even  numbers  from  0  to  200? 

23.  What  is  the  sum  of  the  odd  numbers  from  1  to  200? 

24.  What  is  the  sum  of  the  first  n  even  numbers  ? 

25.  What  is  the  sum  of  the  first  n  odd  numbers  ? 

26.  Insert  9  arithmetical  means  between  ■— |-  and  +|-. 

27.  Sum  the  series  l/|-+l/2+3l/|-+.  .  .  to  20  terms^ 

28.  Sum  the  series  5— 2— 9—.  ..to  8  terms. 

29.  Insert  5  arithmetical  means  between  10  and  8. 

30.  Insert  4  arithmetical  means  between  —2  and  —16. 

31.  Sum  (a  +  dy  +  (a^-\-d'')  +  (a-5y  to  n  terms. 

32.  Find  the  sum  of  the  first  10  multiples  of  3. 

33.  Find  the  sum  of  the  first  50  multiples  of  7. 

34.  Find  the  sum  of  the  odd  numbers  between  200 
and  300. 

35.  The  sum  of  25  successive  terms  of  the  progression 
5_|.8  +  11  +  .  .  .  is  1025  ;   what  is  the  first  term? 

36.  The  sum  of  10  terms  of  an  arithmetical  progression 
is  15,  and  the  fifth  term  is  0 ;  what  is  the  first  term  ? 

37.  How  many  terms  of  the  progression  9  +  13  +  17  +  .  » 
must  be  taken  in  order  that  the  sum  may  equal  624  ? 

38.  Find  the  arithmetical  progression  whose  sum  is  500, 
whose  middle  term  equals  50,  and  whose  last  term  is 
three  times  the  first  term. 

39.  We  must  take  how  many  terms  of  the  progression 
^l+-^W(l+-^W(l+-^W.  .  .  in  order  that  the  sum 
may  be  6^. 


PROGRESSIONS.  331 

40.  How  many  terms  must  be  taken  from  the  com- 
mencement of  the  series  1-f  54-9  +  13  +  17,  etc.,  so  that 
the  sum  of  the  13  succeeding  terms  shall  be  741? 

41.  The  sum  of  the  first  three  terms  of  an  arithmetical 
progression  is  15,  and  the  sum  of  their  squares  is  83; 
find  the  common  difference. 

Let  ;t:=first  term  and;K  the  common  difference.    Then 
a'+(^-f-j/)  +  (^+27)z=15, 
and  ^2  +  (x+7)2  +  (:r+2y)2=z:83. 

Another  notation  which  is  very  convenient  in  a  problem  like  this 
is:  Represent  the  three  terms  by  x—y,  x,  and  x-\-y,  whence  we 
write  {x—y)   +^   -l-(-^+7)   =15, 

and  (^-/)^-4-:*:2_|.(;^^^)2^83^ 

42.  There  are  two  arithmetical  progressions  which  have 
the  same  common  difference ;  the  first  terms  are  3  and  5 
respectively,  and  the  sum  of  seven  terms  of  the  one  is  to 
the  sum  of  seven  terms  of  the  other  as  2  to  3.  Determine 
the  progressions. 

43.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  12,  and  the  sum  of  their  squares  is  QQ>.  Find 
the  numbers. 

44.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  33,  and  the  sum  of  their  squares  is  461.  What 
are  the  numbers? 

GKOMKTRICAI.  PROGRESSIONS. 

495.  A  Geometrical  Progression  is  a  series  of  terms 
such  that  each  term  is  the  product  of  the  preceding  term 
by  a  fixed  factor  called  the  Ratio.  The  following  are 
examples : 

(1)  3,  6,  12,  24,  48.  (3)  i,  i,  i,  -^,  ^\. 

(2)  100,  ~60,  25,  -12f      (4)  a,  ar,  ar\  ar\  ar^. 
The  first  and  last  terms  are  often  called  the  Extremes 

and  the  other  terms  the  Means. 


332  UNIVERSITY    ALGEBRA. 

496.  The  Geometrical  Mean  of  two  numbers  a  and  b 
is  found  as  follows  :  Let  G  stand  for  the  required  mean. 
Then,  by  the  definition  of  a  geometrical  progression, 

G^b_ 

a     G 
Whence,  G^=ab, 

or  G=\/ab.  [4] 

Thus  4  is  the  geometrical  mean  of  2  and  8.  The  arithmetical  mean 
Q)i  2  and  8  is  5.  By  the  geometrical  mean  of  n  positive  numbers  is 
meant  the  positive  value  of  the  n  th  root  of  their  product.  Thus  the 
geometrical  mean  of  8,  9,  and  24  is  12=1^8X9X24. 

497.  The  n  th  Term.  Let  a  represent  the  first  term 
of  any  geometrical  progression,  and  r  the  ratio.  Then 
the  progression  may  be  written 

No,  of  Term:  1.  2.  3.  4.  5. 
Progression:  a^  ar^  ar*^,  ar^,  ar^^  ,  .  . 
We  notice  that,  by  the  nature  of  the  progression,  every 
time  the  number  of  terms  is  increased  by  1  the  exponent 
ot  r  is  increased  by  1  also,  and  the  exponent  of  r  in  any 
term  is  one  less  than  the  number  of  that  term.  Therefore, 
representing  the  n  th  term  by  /, 

l=ar"-i.  [5] 

498.  The  Sum  of  n  Terms.  Representing  by  ^  the 
sum  of  n  terms  of  any  geometrical  progression,  we  have 

s=a+ar+ar'^-\-ar^  +  ,  .  .+ar''-'^  +  ar''-'^,         (1) 
Multiplying  this  equation  through  by  r,  we  get 

rs=:ar+ar'^  +  ar^+ar^  +  ,  .  ,+ar''-^+ar^,         (2) 

Subtracting  (1)  from  (2),  we  have 

rs — s=ar" — a,  (3) 

Whence  s(r—l)  =  ar*'—a, 

ar--a 
or  ^=~i^=T"  [Pi 


PROGRESSIONS.  333 

Now,  from  [5]  l=ar''~'^.  Therefore,  ar'*=r(^r''~^)  =  r/, 
and  [6]  may  be  written 

-?£f-  [7] 

499.  We  give  a  few  examples  of  the  use  of  formulas 
[5],  [6],  and  [7]. 

(1)  Find  the  7th  term  of  the  progression  4+8+16+ .  .  . 
Here  <^=:4,  r=2,  and  «=7,  whence 

/=4x26=r256, 

(2)  Find  sum  of  6  terms  of  progression  13+1. 3  +  .  13+ 
Here  ^=13,  ^=6,  and  r^^^,  whence 

13X(iy«-13. 

10      ■•■ 

13-13000000     _12999987_ 
that   IS,  ''- 100000- 1000000  ~~900000~~-^^-'*^^^'^- 

(3)  Insert  3  geometrical  means  between  31  and  496. 
Here  ^=31,  /=496,  and  n=D,  whence 

496=:31X^^, 
or  ^4  =  16, 

therefore,  r:=  ±  2. 

Consequently  the  required  means  are  62,  124,  and  248,  or  —62,  +124, 
and  -248. 

500.  The  two  equations 

contain  five  letters.  If  any  two  of  them  are  unknown 
numbers  and  the  values  of  the  other  three  are  given,  the 
value  of  the  two  unknown  numbers  can  be  determined 
by  solving  the  system  of  two  equations.  But  if  r  is  an 
unknown  number,  the  equations  of  the  system  are  of  a 
high  degree,  since  n  is  usually  a  large  number  and 
always  greater  than  2  at  least.  In  this  case  we  shall  be 
unable  to  solve  the  system,  as  it  is  beyond  the  range  of 
Chapter  XVII.   Also,  lin  is  an  unknown  number,  we  shall 


334  UNIVERSITY   ALGEBRA. 

have  an  equation  with  the  unknown  number  appearing 
as  an  expoiient,  which  is  a  kind  of  equation  we  have  not 
yet  considered.  Hence  there  are  but  a  limited  number 
of  cases  in  which,  with  our  present  means,  we  can  solve 
the  above  system.  We  give  a  few  samples  of  cases  readily 
solved. 

(1)  Find  the  sum  of  a  geometrical  progression  of  7  terms,  of  which 
the  last  term  is  128,  the  ratio  being  2. 

Here  /==128,  r—2,  and  w=7,  whence 

128=^  2«  (1) 

s= J (2) 

From  {1)  a=2,  whence  from  (2)  j=254, 

(2)  Find  the  sum  of  a  geometrical  progression  of  5  terms,  the  ex- 
tremes being  8  and  10368. 

Here  a=8,  /=  10368,  and  n=:5,  whence 

10368=:8r4  (1) 

r  10368 -8 
'^-       r-1  (2) 

From  (1)  r=Q,  whence  from  (2)  ^=12450. 

(3)  Find  the  extremes  of  a  geometrical  progression  whose  sum  is 
635,  if  the  ratio  is  2  and  the  number  of  terms  7. 

Here  j=635,  r=2,  and  n=.7,  whence 

/=a2^  (1) 

2/-« 
635=— J-  (2) 

Substituting  /  from  (1)  in  (2),  we  get 

635=128^ -d5. 
Whence  a=^;  hence  /=320. 

(4)  The  4th  term  of  a  geometrical  progression  is  4,  and  the  6th 
term  is  1.     What  is  the  10th  term? 

Here  ar^=4:  (1) 

and  ar^  =  l  (2) 

Whence,  by  dividing  (2)  by  (1), 

r«=}. 
Whence,  ^=±|. 

Therefore,  from  (1)  fl=— =±33. 

Then  the  10th  term  is  ±S2{±i)*=^. 


PROGRESSIONS.  335 

EXAMPLKS   AND    PROBIvEMS. 

1.  Find  the  sum  of  7  terms  of  4  +  8  +  16  +  .  .  . 

2.  Find  the  sum  of  9  terms  of  2  +  6  +  18  +  .  .  . 

3.  Find  the  sum  of  7  terms  of  1+4  +  16+ .  .  . 

4.  Find  the  10th  term  and  the  sum  of  10  terms  of 
4-2  +  1-.  .  .  _         _        _ 

5.  Sum  the  series  l"' 3+ 1^6+ 1^12+.  .  .  to  8  terms. 

6.  Sum  the  series  —4+8—16  +  32—.  .  .  to  6  terms. 
'j,S\xma  +  a(l-{-x)  +  a(l+xy  +  .  .  .  to  8  terms. 

8.  Given  /=  78125,  r=5,  n=S;  find  a  and  s, 

9.  Given  /=  2T»  ^— i^  n=6  ;  find  a  and  s, 

10.  Given  5=  635,  n=7,  r=2 ;  find  a  and  /. 

11.  Find  rand  s;  given  a=2\  /=31250,  n=7. 

12.  Find  r  and  s;  given  a=S6,  /=^,  n=7, 

13.  Find  r  and  s;  given  a=3,  /=49152,  n=8. 

14.  Find  r  and  s;  given  a=7,  /=3584,  ;^=10. 

15.  Insert  2  geometrical  means  between  47  and  1269. 

16.  Insert  3  geometrical  means  between  2  and  3. 

17.  Insert  1  geometrical  mean  between  14  and  686. 

18.  Given  a=\,  /=1024,  71=14;  find  rand  s. 

19.  Insert  7  geometrical  means  between  a^  and  d^, 
^o.  Find  the  sum  of  the  first  10  consecutive  powers  of  2. 

21.  Find  the  sum  of  the  first  lOconsecutivepowersof—^-. 

22.  Sum  d(l+xy-^-JrK^+xy-^  +  ,  .  .  to  n  terms. 

23.  Sum  x"-'^  +x*'-^y-j-x"-^j/^  +x*'~^j/^  +  ,  .ton  terms. 

24.  Sum  x"-^  —x^-^y+x^'-'y^  —x"-^y^  +  ..ton  terms. 

25.  Sum  a—ar'^+ar'^'-'ar~^  +  ,  ,  .  to  n  terms. 

26.  Select  6  terms  from  the  progression  ^—2+8—.  .  . 
whose  sum  shall  equal  —6536. 


336  UNIVERSITY    ALGEBRA. 

27.  The  sum  of  the  extremes  of  a  geometrical  pro- 
gression of  4  terms  is  56,  and  the  sum  of  the  means 
is  24.     Find  the  4  terms. 

1      3      5       1       3       5 

28.  Sum  the  series   2  +  22"^23"*"2^"^P~^2«  "*"'  '  '  ^^  ^ 

.     /I  ,   3   ,   5\     /I  ,   3    ,   5\1    , 
terms,  or  the  progression  [2+2^  +  2  3  j  +  ^  2  "*"  22  "^  2  ^  72  ^  + 

to  three  terms. 

29.  The  4th  term  of  a  geometrical  progression  is  192 
.  and  the  7th  term  is  12288 ;  find  the  sum  of  the  first  3 

terms. 

30.  The  6th  term  of  a  geometrical  progression  is  156, 
and  the  8th  term  is  7644  ;  what  is  the  4th  term? 

31.  Prove  that  if  numbers  are  in  geometrical  progres- 
sion their  difierences  are  also  in  geometrical  progression, 
having  the  same  common  ratio  as  before. 

32.  If  a-j-d+c-i-d-}-.  .  .  is  a  geometrical  progression, 
prove  that  (a'  +  d''')-\-(d''+c^)  +  (ic^+d^)  +  .  .  .  is  also  a 
geometrical  progression. 

33.  A  man  agreed  to  pay  for  the  shoeing  of  his  horse 
as  follows:  .0001  cents  for  the  first  nail,  .0002  cents  for 
the  second  nail,  .0004  cents  for  the  third  nail,  and  so  on 
until  the  8  nails  in  each  shoe  were  paid  for.  How  many- 
dollars  did  he  agree  to  pay?  How  much  did  the  last  nail 
cost  him? 

INFINITE   GEOMKTRICAIv   PROGRESSIONS. 

501.  If  the  ratio  of  a  geometrical  progression  is  a 
proper  fraction,  the  progression  is  said  to  be  Decreasing. 
Thus, 

1,  h  hi  and  -J,  i,  2V.  sV 
are  decreasing  geometrical  progressions. 


PROGRESSIONS.  337 

502.  Limit  of  Sum  as  n  Increases.  If  we  increase 
the  number  of  terms  in  the  decreasing  progression  1,  |-, 
J,  .  .  .  the  sum  of  the  terms  will  never  equal  2,  but  will 
approach  2  as  near  as  we  please.  We  wish  to  show  that 
the  sum  of  every  decreasing  geometrical  progression  ap- 
proaches a  deiSnite  limit  as  the  number  of  terms  increases 
without  limit.     We  know 

'-°^-         (1) 

If  we  suppose  r  a  proper  fraction  and  n  increasing 
without  limit,  we  have  the  case  under  consideration.  As 
n  varies,  both  sides  of  (1)  are  variables,  and  since  they 
are  always  equal,  we  have 

limit  s=  limit  P^^J^^n  (2) 

Since  r  is  a  proper  fraction,  the  term  r**  continually 
approaches  0  as  a  limit  as  n  increases.  Whence,  taking 
the  limit  of  the  right  hand  side  of  (2),  we  may  write 

limit  s=j^'  [8] 

(1)  Find  the  limit  of  J— i  +  ^  — tV  +  -     •  as  » increases  without  limit. 
Here  «=J,  andr=— J.     Whence, 

limit  .=f:^L_^j. 

(2)  Find  the  limit  of  .3333,  ... 
Here  «=^0  and  ^=^q.     Whence, 


limit  s 


1-A~^- 
KXAMPI^KS. 

As  n  increases  without  limit,  find  the  limit  ot  each  of 
the  following : 

1.  9-6+4-.  .  .  5.  .272727  .  .  . 

2.  .279279279  ...  6.  ^--1+^^  ... 

22  — U.  A. 


338  UNIVERSITY   ALGEBRA. 

3.  4+.8+.16+.  .  .  7.  1/8  +  1/4+1/2+. 

a+x    a—x  o     ^^     r      ^     . 


a-;r     ^+^     •  •  •  V3  +  I      1/3  +  3 

9.  Express  the  number  8  as  the  sum  of  an  infinite 
geometrical  progression  whose  second  term  is  2. 

HARMONICAL  PROGRESSIONS. 

503.  A  series  of  numbers  which  are  such  that  their 
reciprocals  form  an  arithmetical  progressiom  are  said  to 
form  an  Harmonical  Progression.  The  following  are 
examples : 

(X)hhhh  (4)  i.  1,-1,  -i. 

(2)  1,  i,  i,  A-  (5)  4,  6,  12. 


x—y    X    x+y  a    a+d    a-\-2d 

■  The  first  and  last  terms  are  called  the  Extremes,  and 
the  other  terms  the  Means. 

504.  Fundamental  Property.  The  difference  between 
the  first  and  second  of  any  three  consecutive  terms  in  harmon- 
ical progression  is  to  the  difference  between  the  second  and 
the  third  as  the  first  is  to  the  third. 

Let  a,  by  and  c  be  the  three  terms  in  harmonical  pro- 
gression.    Then  we  have,  by  definition, 

b     a     c     d 
a—b     b—c 
whence.  -W^-b^- 

>w*.        /.  ^ — b    ab    a 

Therefore,  7 —  =-r  =-♦ 

b — c     be     c 

that  is,  a-h\h—c—a\c.  "1)^ 


PROGRESSIONS.  339 

505.  The  Harmonical  Mean  of  two  numbers  a  and  b 
is  found  as  follows :  I<et  H  stand  for  the  required  mean. 
Then  we  have 

H    a     b     H 

^t.  .•  2      11     a^b 

that  IS,  ^-=-+_=_-. 

Whence,  ^^~dVh  [10] 

96 
Thus  the  harmonical  mean  of  4  and  12  is  TTTo'  or  6.     By  the  har- 
monical  mean   of   several  numbers   is   meant   the   reciprocal  of   the 
arithmetical  mean  of  their  reciprocals.     Thus,  the  harmonical  mean 
of  12,  8,  and  48  is  IS^r. 

506.  Although  harmonical  series  are  of  such  a  simple 
character,  no  expression  can  be  found  for  the  sum  of  n 
terms.  But  our  knowledge  of  arithmetical  progressions 
enables  us  to  find  the  value  ot  any  required  term  of  a 
harmonical  progression  and  to  insert  any  required  number 
of  harmonical  means  between  two  given  extremes,  as  in 
the  examples  below. 

The  fact  of  having  an  expression  in  a  "finite  form"  which  equals  the 
sum  of  a  series  is  an  exceptional  one.  The  fact  that  such  expres- 
sions can  be  found  for  arithmetical  and  geometrical  progressions 
should  be  considered  remarkable,  rather  than  to  deem  it  surprising 
that  no  such  expression  exists  for  a  harmonical  progression. 

(1)  Write  6  terms  of  the  harmonical  progression  6,  3,  2. 

We  must  write  6  terms  of  the  arithmetical  progression  J,  i,  J. 
The  common  difference  in  this  latter  is  \,  so  we  have  the  arithmetical 
progression 

i     1     1      3      5      1 
6»    3»    2'    3'    6'    ^• 

Whence  we  have  the  harmonical  progression 
6,    3,    2.    1.5.     1.2,    1. 

(2)  Insert  2  harmonical  means  between  4  and  2. 

We  must  insert  two  arithmetical  means  between  \  and  J;  these  are 
\  and  ^^,     Whence  the  required  harmonical  means  are  3  and  2.4. 


340  UNIVERSITY   ALGEBRA. 

507.  Relative  Values  of  A,  G,  and  H.  We  have 
found  the  following  values  for  the  arithmetical,  geomet- 
rical, and  harmonical  means  of  any  two  positive  numbers* 

From  these  we  find 

A-G=-\(a+b)-^/^b-=\{\/a-Vby,         (1) 

also,         G^H^V7b^^=y^(Va^  VI)  \         (2) 
a-\-b     a-\-b^  ' 

Now,  since  a_and  b  stand  for  positive  numbers,  we 

know  {y a—V b)'^  is  positive,  and  also  a-\-b   and  V ab 

are  positive.  Whence  A—G  and  G—Hqxq  each  positive, 

that  is,  the  arithmetical  mean  of  two  positive  numbers  is 

greater  than  their  geometrical  mean,  and  their  geometrical 

mean  is  greater  than  their  harmonical  mean.     This  may 

be  expressed 

A>G>H.  [10] 

508.  Relation  between  A,  G,  and  H.  The  follow- 
ing relation  is  important.     As  before,  we  have 

Whence,  AH^ab, 

But  G'^=-ab. 


Therefore,  G^V^S.  [11] 

That  is  to  say,  the  geometrical  mean  of  any  two  positive 

numbers  is   the  same  as   the  geometrical  mean   of  their 

arithmetical  and  harmonical  means. 

MISCKI.I.ANEOUS   KXCERCISKS. 

A.  P..  G.  P.,  and  H.  P.  stand  for  Arithmetical^  Geometrical^  and 
Harmonical  Progression,  respectively. 

1.  Continue  the  H.  P.  12,  6,  4. 

2.  Sum  the  series  l  +  2r+3r2+4r^  +  .  .  ,  \.o  n  terms. 

3.  If  a,  b,  Cy  ^be  in  A.  P.,  a,  e,f  din  G.  P.,  a,  g^  h^ 
din  H.  P.,  then  ad=ef=bh=cg. 


PROGRESSIONS.  '  34I 

4.  The  sides  of  a  right  triangle  are  in  A.  P.     Show 
that  they  are  proportional  to  3,  4,  5. 

5.  Three  numbers  are  in  G.  P.;  if  each  be  increased 
by  15  they  are  in  H.  P.     Find  them. 

6.  If  the  sum  of  the  first  p  terms  in  an  A.  P.=0,  the 

sum  of  the  next  q  terms = /  1 

p—1 

7.  If  ^,  b,  c,  dhe  in  H.  P.,  show  that 

8.  If  X,  y,  z  be  in  G.  P. ,  prove  that 

9.  Find  the  difference 

(lf+li+f4-.  .  to5terms)-(l|+li+f+.  .  to5terms). 
ID.  If  the  A.  mean  between   two  numbers  equals  1, 
show  that  the  H.  mean  is  the  square  of  the  G.  mean. 

11.  If  x—a,  y—a,  and  z—a  be  a  G.  P.,  prove  that 
twice  jj'—^^^  is  the  harmonic  mean  between  y — x  and  y—z. 

12.  If  a,  b,  c  be  in  A.  P,  and  x  be  the  G.  mean  of  <3J-and 
b,  and  y  the  G.  mean  of  b  and  c,  then  will  x",  b'^ ,  y'^  be 
in  A.  P. 

13.  Find  an  equation  whose  roots  will  be  the  arith- 
mitical  and  harmonical  means  between  the  roots  of 
x'^—px+q=0. 

14.  The  sum  of  10  terms  of  an  A.  P.  is  145,  and  the 
sum  of  its  fourth  and  ninth  terms  is  5  times  the  third ; 
determine  the  series. 

15.  The  A.  mean  between  two  numbers  is  to  the  G. 
mean  as  5  : 4  and  the  difference  of  their  G.  and  H.  means 
is  34;  find  the  numbers. 

16.  To  each  of  three  consecutive  terms  of  a  G.  P.  the 
second  of  the  three  is  added.  Show  that  the  three 
resulting  numbers  are  in  H.  P. 


342  .  UNIVERSITY   ALGEBRA. 

17.  Show  that  the  product  of  any  odd  number  of  con- 
secutive terms  of  a  G.  P.  will  be  equal  to  the  n  th  power 
of  the  middle  term,  n  being  the  number  of  terms. 

18.  The  natural  numbers  are  divided  into  groups,  as 
follows:  1;  2,  3;  4,  5,  6;  7,  8,  9,  10;  and  so  on.  Prove 
that  the  sum  of  the  numbers  in  the  ^th  group  is  \k{k'^  -|- 1). 

19.  If  a,  by  Cy  X  be  all  real  numbers  in  the  equation 

prove  that  a,  b^  c  are  in  G.  P.  and  that  x  is  their  common 
ratio. 

20.  AG.  P.,  whose  common  ratio  is  V«,  has  the  same 
first  and  second  terms  as  an  H.  P.  Prove  that  the  third 
term  of  the  former  series  will  be  equal  tothe(«+2)nd 
term  of  the  latter. 

21.  If  ^1,  ^2»  -^s)  ^tc,  are  the  sums  of  r  A.  P.,  each  to  n 
terms,  the  first  terms  being  1,  2,  3,  etc.,  respectively,  and 
the  difierences  1,  3,  5,  etc.,  respectively,  show  that 

^i+^2+'y8  +  -  •  •  '\'Sr=\nr{rir-\-V). 


CHAPTER  XXI. 

ARRANGEMENTS   AND   GROUPS. 

509.  Every  different  order  in  which  given  things  can 
be  placed  is  called  an  Arrangement  or  Permutation, 
and  every  different  selection  that  can  be  made  is  called  a 
Group  or  Combination. 

Thus  if  we  take  the  letters  a,  by  Cy  two  at  a  time,  there 
are  six  arrangements,  viz. : 

aby  ac,  ba,  be,  ca,  cby 
but  there  are  only  three  groups,  viz. : 
ab,  ac,  be. 
If  we  take  the  letters  a,  b,  e  all  at  a  time,  there  are 
six  arrangements,  viz.: 

abe,  aeb,  bae,  bca,  eaby  ebUy 
but  there  is  only  one  group,  viz. : 

abe. 

510.  Theorem.  The  number  of  arrangements  of  n 
different  things  taken  all  at  a  time  equals  n  times  the  num- 
ber of  arrangements  ofn—1  things  taken  all  at  a  time. 

Let  us  first  take  two  special  cases  and  then  pass  to  the 
general  case. 

First.  If  we  take  three  things,  say  «,  3,  Cy  there  are  six 
arrangements,  viz.: 

abCy  acby  bac,  bca,  cab,  cba. 
Now  these  six   arrangements   may  ho,  looked  upon  as 
composed  of  three  classes.     In  the  first  class  the  letter  a 
stands  first,  in  the  second  class  the  letter  b  stands  first, 
and  in  the  third  class  the  letter  c  stands  first.     The 


344  UNIVERSITY   ALGEBRA. 

arrangements  in  the  first  class  may  be  looked  upon  as 
obtained  by  arranging  the  two  letters  b  and  c  in  every 
possible  way,  and  writing  a  before  each  arrangement; 
those  in  the  second  class  may  be  obtained  by  arranging 
the  letters  a  and  c  in  every  possible  way  and  writing  b 
before  each  arrangement ;  those  in  the  third  class  may  be 
obtained  by  arranging  the  letters  a  and  b  in  every  possible 
way  and   writing  c  before  each  arrangement. 

By  considering  the  arrangements  formed  in  the  manner 
just  described  it  is  evident  that  all  three  letters  appear  in 
each  arrangement,  and  it  is  also  evident  that  the  number  of 
arrangements  in  which  a  stands  first  is  exactly  the  same 
as  the  number  in  which  b  stands  first,  and  also  exactly 
the  same  as  the  number  in  which  c  stands  first.  Hence, 
we  see  that  there  are  three  times  as  many  arrangements 
of  three  things  taken  all  at  a  time  as  there  are  of  two 
things  taken  all  at  a  time. 

Second,  If  we  take  four  things,  say  a,  b,  c,  d,  then  we 
may  arrange  the  three  letters  b,  c,  d  in  every  possible  way 
and  place  a  before  each  arrangement,  then  arrange  the 
three  letters  a,  a  d  \vl  every  possible  way  and  place  b 
before  each  arrangement,  then  arrange  the  three  letters 
a,  b,  d  in  every  possible  way  and  place  the  letter  c  before 
each  arrangement,  and  finally  arrange  the  three  letters 
a,  b,  ^  in  every  possible  way  and  place  the  letter  d  before 
each  arrangement.  It  is  evident  that  all  four  letters  a,  b, 
c,  d  appear  in  each  arrangement  thus  formed,  and  it  is 
also  evident  that  the  number  of  arrangements  in  which  a 
stands  first  is  exactly  the  same  as  the  number  in  which  b 
stands  first,  and  so  bn. 

Hence  there  are  in  all  four  times  as  many  arrangements 
of  four  things  taken  all  at  a  time  as  there  are  ol  three 
things  taken  all  at  a  time. 


ARRANGEMENTS  AND  GROUPS.  345 

In  general,  if  we  have  n  things,  say  the  letters  a,  b,  c, 
d,  e,  f,  .  .  then  we ,  may  suppose  all  the  letters  but  a 
arranged  in  every  possible  order  and  then  a  placed  before 
each  of  these  arrangements ;  next  we  may  suppose  all  the 
letters  but  b  arranged  in  every  possible  order  and  then 
b  placed  before  each  of  these  arrangements,  and  so  on. 

It  is  evident  that  all  n  letters  appear  in  each  arrange- 
ment thus  formed,  and  it  is  also  evident  that  the  number 
of  arrangements  in  which  a  stands  first  is  exactly  the 
same  as  the  number  in  which  any  other  letter  stands  first. 

Now  the  number  of  arrangements  in  which  a  stands 
first  is  evidently  the  number  of  arrangements  of  {n—V) 
things  taken  all  at  a  time,  and  hence  the  total  number  of 
arrangements  of  n  things  taken  all  at  a  time  is  n  times 
the  number  of  arrangements  oi  n—\  things  taken  all  at 
a  time. 

Let  us  represent  the  number  of  arrangements  oin  things 
taken  all  at  a  time  by  A^.  Then,  by  what  has  just  been 
shown  we  have  the  formula 

KXAMPI.BS. 

1.  If  the  number  of  arrangements  of  9  things  taken  all 
at  a  time  equals  x,  the  number  of  arrangements  of  10 
things  taken  all  at  a  time  equals  what? 

2.  Using  the  result  found  in  example  1  for  the  number 
of  arrangements  of  10  things  all  at  a  time,  find  the  num- 
ber of  arrangements  of  11  things  taken  all  at  a  time. 

3.  If  ^6=;r,  what  does  Aq  equal? 

4.  If  ^7  =  10;ir,  what  does  A^  equal? 

5.  Express  A12  ^^  terms  of  ^  ^  ^ . 

6.  Express  ^  5  q  in  terms  of  ^  4  9 . 

7.  Express  A^  in  terms  of  A^_^, 


346  UNIVERSITY   ALGEBRA. 

511.  Problem.  To  find  the  number  of  arrangements 
ofn  different  things  taken  all  at  a  tiifie. 

By  the  theorem  of  the  last  article  we  may  write  each 
of  the  following  equations  except  the  last  one : 
A„^=nA„_iy 
A„_^  =  (n--1)A„_2, 

A  ^= 6  A  2 , 

A,  =  l. 

The  last  equation  of  this  list  is  not  written  by  an  appli- 
cation of  the  theorem,  but  is  evidently  true,  for  one  thing 
can  evidently  be  arranged  in  but  one  way. 

Now  multiplying  these  equations  together,  member  by 
member,  we  get 

A^A^A^,..  A„=2A^  3^2  •  •  •  ^-^^-i 

=  lx2x3...;^^l^2•••  ^«-i- 
By  cancelling  the  common  factors,  we  get 
An=lX2xS     .  .  n. 
The  product  of  the  integer  numbers  from  n  down  to  1 
or  from  1  up  to  «  is  often  represented  by  in  or  nl  ^  and  is 
read  factorial  n^  or  n  admiration. 
With  this  notation  we  may  write 

EXAMPI^BS. 

1.  How  many  arrangements  can  be  made  of  5  things 
taken  all  at  a  time  ? 

2.  How  many  different  numbers  can  be  made  with  the 
four  digits  1,  2,  3,  4,  using  each  digit  once  and  only  once 
to  form  each  number? 

3.  The  number  of  arrangements  of  four  things  taken 
all  at  a  time  bears  what  ratio  to  the  number  of  arrange- 
ments of  six  things  taken  all  at  a  time  ? 


ARRANGEMENTS  AND  GROUPS.  34/ 

4.  How  many  four-figure  numbers  can  be  formed  with 
the  digits  1,  2,  3,  4,  having  1  for  the  first  digit  in  each 
number  and  using  each  digit  once  and  only  once  in  each 
number? 

5.  How  many  five-figure  numbers  can  be  formed  with 
the  digits  1,  2,  3,  4,  5,  having  1  for  the  first  and  5  for  the 
last  digit  in  each  number,  and  using  each  digit  once  and 
only  once  in  each  number? 

512.  Theorem.  The  number  of  arrangements  of  n 
different  things  taken  r  at  a  time  is  equal  to  n  times  the 
number  of  arrangements  of  n—\  things  taken  r — 1  at  a 
time. 

Let  us  first  take  a  particular  case,  say  the  number  of 
arrangements  of  five  things,  say  the  five  letters  a,  b,  c,  d,  <?, 
taken  three  at  a  time.  Suppose  the  arrangements  all 
made  and  we  select  those  which  begin  with  a  and  put 
them  by  themselves  in  one  "class,  then  those  which  begin 
with  b  and  put  them  by  themselves  in  another  class,  and 
so  on.  We  then  divide  the  whole  number  of  arrange- 
ments into  five  classes,  and  it  is  evident  that  the  number 
in  any  one  class  is  just  the  same  as  the  number  in  any 
other  class.  Consider  those  which  begin  with  a.  Then 
every  arrangement  in  this  class  contains,  besides  a,  two  of 
the  four  letters  by  c,  d,  e,  and  since  a  is  fixed  and  the 
other  letters  are  arranged  in  every  possible  way,  therefore 
the  number  of  these  arrangements  must  equal  the  number 
of  arrangements  of  the  four  letters  ^,  c,  d,  e  taken  two  at 
a  time. 

In  general,  if  we  have  n  things,  say  the  letters  a,  b,  c, 
d,  e,f,.  to  be  taken  r  at  a  time,  we  may  select  all  those 
arrangements  which  begin  with  a  and  put  them  by  them- 
selves in  one  class,  then  those  which  begin  with  b  and 
put  them  by  themselves  in  another  class,  and  so  on.   We 


348  UNIVERSITY    ALGEBRA. 

thus  divide  the  whole  number  of  arrangements  into  n 
classes,  and  it  is  evident  that  the  number  of  arrangements 
in  any  one  class  is  just  the  same  as  the  number  of 
arrangements  in  any  other  class. 

Consider  those  which  begin  with  a. 

Then  every  arrangement  in  this  class  contains  besides 
a,  {r—V)  of  the  letters  b,  c,  d,  .  .  ,  and  since  a  is  fixed 
while  the  remaining  letters  are  arranged  in  every  possible 
order,  therefore  the  number  of  arrangements  in  the  class 
considered  must  equal  the  number  of  arrangements  of 
n—\  letters  b,  c,  d,  .  .  ,  taken  r— 1  at  a  time. 

As  there  are  n  such  classes  and  as  the  number  of 
arrangements  in  each  class  equals  the  number  of  arrange- 
ments of  ^  — 1  things  taken  r—l  at  a  time,  therefore  the 
total  number  of  arrangements  of  n  things  taken  r  at  a 
time  equals  n  times  the  number  of  arrangements  oi  n  —  1 
things  taken  r—l  at  a  time. 

Let  us  represent  the  number  of  arrangements  of  n  things 
taken  r  at  a  time  by  AQi),  then  by  what  has  just  been 
proved,  we  have  the  formula 

KXAMPIvKS. 

1.  If  the  number  of  arrangements  of  9  things  taken  6 
at  a  time  equals  x,  the  number  of  arrangements  of  10 
things  taken  7  at  a  time  equals  what? 

2.  Using  the  result  found  in  example  1  for  the  number 
of  arrangements  of  10  things  taken  7  at  a  time,  find  the 
number  of  arrangements  of  11  things  taken  8  at  a  time. 

3.  li  A{X)==x,  what  does  A(J)  equal? 

4.  1{aIx)  =  ^x,  what  does  A(J)  equal? 

5.  Express  ^(1)  in  terms  of  y4(|). 

6.  Express  aIi  %)  in  terms  of  a\i  |). 

7.  Express  A(Z)  in  terms  of  y^CzJ). 


ARRANGEMENTS  AND  GROUPS.  349 

513.     Problem.      To  find  the  number  of  at  rang  ements 
of  n  different  things  taken  r  at  a  time^  r  being  less  than  n. 
By  the  theorem  of  the  last  article  we  may  write  each  of 
the  following  equations  except  the  last  one : 
AC;)^nAC^Zi). 
^(;Z1)=(;^-1MCZ|). 


^(«-''+2)=(;e-^+2)^Q-''+i). 

The  last  equation  of  this  list  is  not  written  by  an  appli- 
cation of  the  theorem  of  the  preceding  article,  but  it  is 
evident  that  the  number  of  arrangements  of  any  number 
of  things  taken  one  at  a  time  equals  the  number  of  things. 

An  inspection  of  the  equations  just  written  shows  that 
in  each  equation  the  number  of  things  taken  at  a  time  is 
one  less  than  in  the  preceding  case,  and  it  is  plain  that 
the  last  equation  of  the  list  which  can  be  written  which 
has  any  meaning,  is  on^  which  gives  the  number  of 
arrangements  of  a  certain  number  of  things  taken  one  at 
a  time. 

Now  multiplying  the  above  equations  together  member 
by  member  and  cancelling  common  factors,  we  obtain 

This  result  may  be  put  in  another  form  which  is  some- 
times more  convenient  than  the  form  here  given. 

Multiplying  and  then  dividing  the  right-hand  member 
by  {n—r){n—r--X)  ....  3.  2.  1,  we  obtain 
4r^.^     ^0^^1)(^-2)  '  ■  (7^~r+l)(^-r)(/^~r~l)  .  .  1 

^"^  {n-r^in-r-X)  ...  3.  2.  1. 


3  so  UNIVERSITY   ALGEBRA. 

It  is  easily  seen  that  the  numerator  is   h  and  the 
denominator  is  |;g— r,  hence 

^^yrj—r—-' 


EXAMPLKS. 

1.  How  many  arrangements  can  be  made  of  8  things 
taken  3  at  a  time? 

2.  How  many  arrangements  can  be  made  of  8  things 
taken  5  at  a  time? 

3.  How  many  three-figure  numbers  can  be  formed 
with  the  six  digits  1,  2,  3,  4,  5,  6,  without  repeating  any 
digit  in  any  number  ? 

4.  How  many  four-figure  numbers  can  be  formed  with 
the  six  digits  1,  2,  3,  4,  5,  6,  without  repeating  any  digit 
in  any  number  ? 

5.  How  many  two-figure  numbers  can  be  formed  with 
the  six  digits  1,  2,  3,  4,  5,  6,  without  repeating  any  digit 
in  any  number? 

6.  How  many  three-figure  numbers  between  100  and 
200  can  be  formed  with  the  five  digits  1,  2,  3,  4,  5,  with- 
out repeating  any  digit  in  any  number? 

514.  Problem.  To  find  the  number  of  groups  of  n 
different  things  taken  r  at  a  time. 

Take  the  letters  a,  b,  c,  ^,  <?,...  ,  and  suppose  the 
groups  all  written  down;  then,  fixing  our  attention  upon 
any  one  group,  it  is  evident  that  there  could  be  several 
different  arrangements  made  from  that  group  by  changing 
the  order  of  the  letters. 

It  is  further  evident  that  if  we  form  all  possible  arrange- 
ments in  each  group  we  thereby  obtain  the  total  number 
of  arrangements  of  the  n  letters  taken  /-  at  a  time. 


ARRANGEMENTS  AND  GROUPS.  35  I 

The  total  number  of  arrangements  then  equals  the 

number  of  arrangements  in  each  group  multiplied  by  the 

number   of  groups.     Hence,    representing   the  number 

of  groups  of  n  things  taken  r  at  a.  time  by  G^f)  and 

remembering  that  the  number  of  arrangements  in  each 

group  equals  the  number  of  arrangements  of  r  things 

taken  all  at  a  time,  that  is  Ir,  and  further  remembering 

\n 
that  the  total  number  of  arrangements  equals  y-!= — 

\n 
we  have  \r  G(f)=-r-^=^ 

hence  <^(r)  = 


|r    \n—r 

EXAMPLES. 

1.  How  many  groups  can  be  made  of  8  things  taken  6 
at  a  time  ? 

2.  How  many  groups  can  be  made  of  8  things  taken  2 
at  a  time  ? 

3.  How  many  products  can  be  made  from  the  letters 
a,  d,  c,  d,  e,  by  taking  2  of  these  letters  to  form  each 
product  ? 

4.  How  many  products  can  be  made  from  the  letters 
a,  b,  c,  d,  e,  by  taking  3  of  these  letters  to  form  each 
product? 

5.  How  many  products  can  be  made  from  12  different 
letters  by  taking  4  letters  to  form  each  product? 

6.  How  many  products  can  be  made  from  12  different 
letters  by  taking  8  letters  to  form  each  product  ? 

515,  The  form  of  this  result  shows  that  the  number  of 
groups  of  n  things  taken  r  at  a  time  is  the  same  as  the 


352  UNIVERSITY   ALGEBRA. 

number  taken  n—r  at  a  time.  This  is  also  evident  in 
another  way,  for  every  time  we  select  r  things  from  n 
things  we  leave  out  n—r  things ;  hence  there  must  be  as 
many  ways  o£  leaving  out  n—r  things  as  of  selecting  r 
things,  but  of  course  there  are  as  many  ways  of  selecting 
n—r  things  as  there  are  of  leaving  out  n — r  things. 

516.  In  all  that  precedes,  it  was  supposed  that  the 
given  things  were  all  different  and  that  in  forming  the 
arrangements  or  groups  none  of  the  given  things  were 
repeated.  Let  us  now  consider  arrangements  and  groups 
in  which  the  things  may  be  repeated  and  those  in  which 
the  given  things  are  not  all  alike. 

517.  Problem.  To  find  the  number  of  arrangements 
of  n  things  taken  r  at  a  time,  repetitions  being  allowed. 

Suppose  firstnwe  wish  the  number  of  arrangements, 
including  repetitions,  of  the  four  letters  a,  b,  c,  d  taken 
one  at  a  time.  Evidently  there  are  four  arrangements, 
viz.:  «,  b,  c,  d. 

Next  suppose  we  wish   the   arrangements,  including 
repetitions,  of  the  four  letters  a,  b,  :,  d  taken  two  at  a  time. 
The  arrangements  are  the  following: 
aa  ab  ac  ad 
ba  bb  be  bd 
ca   cb   cc  cd 
da  db  dc  dd 
Thus  we  see  that  there  are  sixteen  arrangements,  that 
is,  42  arrangements.  In  exactly  the  same  way  if  we  have 
n  letters  a,  b,  c,  d,  e,f  ,  .  ,  ,  the  a  may  be  followed  by 
each  of  the  n  letters,  giving  n  arrangements  beginning 
with  a ;  the  b  may  be  fellowed  by  each  of  the  n  letters, 
giving  n  arrangements  beginning  with  b,  etc.     So  there 
are  7t  arrangementssbeginning  with  each  letter;    hence  in 


ARRANGEMENTS  AND  GROUPS.  353 

all  there  are  nP'  arrangements  of  n  things  taken  two  at  a 
time,  allowing  repetitions. 

Let  us  now  find  the  number  of  arrangements,  allowing 
repetitions,  of  n  things  taken  three  at  a  time;  and  first  to 
give  definiteness  to  the  ideas,  consider  the  number  of 
arrangements,  allowing  repetitions,  of  four  letters  a,  b,c,d 
taken  three  at  a  time.  We  have  written  out  the  sixteen 
arrangements  of  four  letters  taken  two  at  a  time,  repeti- 
tions being  allowed,  and  now  we  may  suppose  each  of 
these  sixteen  arrangements  to  be  preceded  by  the  letter  a, 
then  each  of  these  sixteen  arrangements  to  be  preceded 
by  3,  etc.  We  then  have  sixteen  arrangements  of  three 
letters  each,  beginning  with  each  letter,  and  as  there  are 
four  letters  there  are  in  all  four  times  sixteen,  or  sixty-four, 
i.  e,  4:^,  arrangements  of  the  letters  «,  dy  Cy  d  taken  three 
at  a  time,  repetitions  being  allowed. 

Now,  in  the  same  way,  if  we  have  n  letters  «,  by  r, 
^,  ^,  /,...,  we  may  suppose  each  of  the  n'^  arrangements 
two  at  a  time  to  be  preceded  by  a,  then  each  of  the  n^ 
arrangements  to  be  preceded  by  by  etc. 

Thus  we  get  n'^  arrangements  beginning  with  ay  n^ 
arrangements  beginning  with  by  n'^  arrangements  begin- 
ning with  Cy  etc.  Hence  in  all  we  obtain  n  times  ;z^,  or 
n^y  arrangements  of  n  letters  taken  three  at  a  time, 
repetitions  being  allowed. 

In  generaly  if  we  know  the  number  of  arrangementsiof 
n  letters  taken  ^  at  a  time,  repetitions  being  allowed,  we 
may  find  the  number  of  arrangements  of  the  n  letters 
taken  ^+1  at  a  time. 

Representing  the  number  of  arrangements,  with  repeti- 
tions, of  n  letters  ay  by  Cy  dy  e^  fy  ,  .  ,  ,  taken  ^  at  a  time 
by  Ns  we  may  then  write  a  before  each  of  these  Ns 
arrangements  i*  at  a  time  and  obtain  N^  arrangements 

^+1  at  a  time  beginning  with  a, 
23  — u.  A. 


354  UNIVERSITY    ALGEBRA. 

We  may  also  write  b  before  each  of  the  same  N^ 
arrangements  and  obtain  Ns  arrangements  ^+1  at  a  time 
beginning  with  b,  and  so  on  until  each  of  the  n  letters 
a,  b,  c,  d,  ...  is  in  turn  placed  before  each  of  the  N^ 
arrangements  ^  at  a  time,  and  we  thus  obtain  nNs 
arrangements  taken  ^+1  at  a  time,  repetitions  being 
allowed. 

Representing  this  number  by  N^^,  we  have 

s  being  a  positive  integer  which  may  be  greater  or  less 
than  a. 

Giving  ^in  turn  all  intermediate  values  from  r— 1  down 
to  1  and  remembering  that  the  number  of  arrangements 
one  at  a  time  is  equal  to  tz,  we  have 

Multiplying  these  equals  together  and  cancelling  the 
common  factors,  we  get 

Nr=nr. 

518.  Problem.  To  find  the  number  of  groups  of  n 
things  taken  r  at  a  time,  repetitions  being  allowed. 

To  prepare  the  way  for  the  general  case  we  begin  with 
the  groups  of  the  four  letters  a,  b,  c,  d  taken  three  at  a 
time,  repetitions  being  allowed. 

In  this  case  there  are  twenty  groups,  viz. : 
aaa  aab  aac  aad  abb 
abc  abd  ace  acd  add 
bbb   bbc  bbd  bcc    bed 
bdd  ccc    ccd  cdd  ddd 


M 


ARRANGEMENTS  AND  GROUPS.  35  5 

Now  if  in  each  of  these  twenty  groups  we  leave  the  first 

letter  standing  and  advance  the  second  letter  one  step  and 

the  third  letter  two  steps,  we  get  twenty  new  groups  of 

the  six  letters  a,  b,  c,  d,  e,  /,  as  follows : 

ahc  abd  abe   abf  acd 

ace  acf   ade   ad/  aef 

bed  bee     bcf    bde  bdf 

bef  cde    cdf    eef  def 

The  groups  here  written  are  the  groups  of  the  six 
letters  a,  b,  c,  d,  e,  f^  without  repetitions. 

In  a  similar  manner  we  may  deal  with  the  general  case 
of  the  number  of  groups  of  n  letters  a,  b,  c,  d,  e,  f,  .  .  . 
taken  r  at  a,  time,  repetitions  being  allowed.  Let  the 
number  of  these  groups  be  denoted  by  n^  and  suppose 
them  all  written  down  in  alphabetical  order  ;  then  in  each 
of  these  groups  keep  the  first  letter  unchanged,  advance 
the  second  letter  one  step,  the  third  letter  two  steps,  the 
fourth  letter  three  steps,  and  so  on. 

We  thus  form  n^  new  groups  containing  all  the  letters 
the  original  ones  contained,  and  r— 1  other  letters. 
These  new  groups  are  written  in  alphabetical  order, 
because  the  original  ones  were,  and  by  the  way  in  which 
the  letters  have  been  advanced  it  is  evident  that  no  letter 
is  repeated  in  any  of  these  new  groups. 

No  two  of  these  new  groups  are  alike,  else  two  of  the 
original  groups  would  have  been  alike. 

Now  since  each  of  these  new  groups  contain  r-f  1  of 
the  n+r—1  letters  a,  b,  c,  d,  e,  .  .  .  ,  and  since  no  letter 
is  repeated  in  any  group,  and  since  no  two  groups  are 
alike,  therefore  these  new  groups  constitute  some  or  all 
of  the  groups  of  the  n-\-r—\  letters  a,  b,  c,  d,  e,  .  .  . 
taken  r  at  a  time  without  repetitions. 

Let  the  number  of  groups  without  repetitions  oin  +  r—  1 
things  taken  r  at  a  time  be  represented  by  6^(""^''~^),  then 


3S6  UNIVERSITY   ALGEBRA. 

it  is  evident  that  n^  cannot  exceed  6^(""*'''~^).  Now  let  us 
conceive  each  of  the  6^(?'*''*""^)  groups  written  down  in 
alphabetical  order,  and  then  leaving  the  first  letter  in 
each  group  unchanged,  change  the  second  letter  in  each 
group  to  the  one  just  before  it  in  the  alphabet,  the  third 
one  in  each  group  to  the  second  one  before  it  in  the 
alphabet,  and  so  on  ;  then  these  groups  are  changed  into 
new  groups  wherein  some  of  the  letters  are  repeated,  and 
no  letter  is  beyond  the  n\h  letter  of  the  alphabet,  More- 
over no  two  of  these  groups  are  alike,  since  no  two  from 
which  they  were  formed  were  alike,  so  that  these  new 
groups  must  be  some  or  all  of  the  groups  of  n  letters  a,  b^ 
c,  d,  e,  .  .  .  taken  r  at  a  time  with  repetitions. 

These  last  formed  groups  are  6^(?"^''~^)  in  number, 
being  formed  from  that  number  of  groups.,  and  as  the 
number  of  groups  with  repetitions  of  n  things  taken  r  at 
a  time  has  already  been  represented  by  n^,  hence  6^(?"^''~^) 
cannot  exceed  n^. 

It  was  previously  proved  that  n^  could  not  exceed 
Q^j^r-i-^^  hence,  since  neither  can  exceed  the  other,  the 
number  must  be  the  same,  or,  in  other  words,  the  num- 
ber of  groups  of  n  things  taken  r  at  a  time,  repetitions 
being  allowed,  is  equal  to  the  number  of  groups  of 
(n  +  r—V)  things  taken  r  at  a  time  without  repetitions. 
The  last  number  has  already  been  found.  Hence  the 
number  of  groups  of  n  things  taking  r  at  a  time,  repeti- 
tions being  allowed,  equals 

{n-\-r-l){n+r—2)  ,  .  .  n 
\r 

which  may  be  written  in  either  of  the  forms 

n{n+l)  .  .  .  jn+r-l) 

\r 

\n+r—l 


ARRANGEMENTS  AND  GROUPS.  35/ 

519.  Problem.  To  find  the  number  of  arrangements 
when  the  given  things  are  not  all  different. 

Illustration. — From  what  has  gone  before  we  know 

that  the  number  of  arrangements  of  the  letters  a,  b,  c,  d, 

taken  all  at  a  time  is  twenty-four,  but  if  we  have  the 

letters  a,  a,  b,  c  the  number  of  arrangements  is  only 

twelve.     These  twelve  are  the  following  : 

aabc  aacb  abac  abca 

acab  acba  baac  baca 

bcaa  caab  caba  cbaa 

If  we  have  the  letters  a,  a,  b,  b  there  are  only  six 
arrangements,  viz.: 

aabb  abab  abba 
baab  baba  bbaa 

If  we  have  the  letters  a,  a,  a,  b  there  are  only  four 
arrangements,  viz.: 

aaab  aaba  abaa  baaa. 

Thus  we  see  that  with  a  given  number  of  things  the 
number  of  arrangements  depends  upon  how  many  of 
each  kind  are  alike. 

Suppose  now  we  have  in  all  n  letters,  of  which  a  is 
repeated  r  times,  b  is  repeated  s  times,  c  is  repeated  t 
times,  and  so  on,  so  that  r+^+/-f .  .  .  =«,  and  we  wish 
to  find  the  number  of  arrangements  taking  all  the  n  letters 
at  a  time. 

Fixing  our  attention  upon  any  arrangement  whatever 
of  the  n  letters,  let  all  the  letters  but  the  a's  remain  un- 
changed while  the  r  a's  change  places  among  themselves. 
Because  all  these  ^'s  are  alike  we  get  only  one  arrange- 
ment, but  if  they  had  all  been  different  we  would  have 
obtained  \r  arrangements,  and  since  the  same  thing  is 
true  whatever  the  arrangements  upon  which  we  fixed 
our  attention  to  begin  with,  it  follows  that  there  are  \r 
times  as  many  arrangements  when  all  the  r  letters  are 


358  UNIVERSITY    ALGEBRA. 

different  as  there  are  under  the  present  supposition.  In 
the  same  way  there  are  \s  times  as  many  arrangements 
when  the  s  d's  are  all  different  as  there  are  under  the 
present  supposition  ;  also,  there  are  1/  times  as  many 
arrangements  when  the  /f  ^s  are  all  different  as  there  are 
under  the  present  supposition,  and  so  on. 

Hence  there  are  Ir  U  1^  .  .  .  times  as  many  arrangements 
when  the  n  letters  are  all  different  as  there  are  under  the 
present  supposition,  or  the  number  of  arrangements  under 
the  present  supposition  is  equal  to  the  number  of  arrange- 
ments of  n  things  taken  all  at  a  time,  when  all  are 
different,  divided  by  Ir  [^  1^  .  .  ,  that  is,  the  number  of 
arrangements  under  the  present  supposition  is  equal  to 

520.  Problem.  To  find  the  number  of  zvays  in  whick 
n  things,  no  two  alike,  can  be  made  up  into  sets  of  which 
the  first  set  contains  r  things^  the  second  set  contains  s  things  y 
the  third  set  contains  t  things^  and  so  on,  where  of  course 
r-\-s-\-t-\-  .  .  .  ^=n. 
We  begin  with  a  special  case  and  find  the  number  of 
ways  in  which  five  letters  a,  b,  c,  d,  e,  can  be  made  up  into 
two  sets  of  which  the  first  set  contains  two,  and  the  second 
set  three  letters. 

Consider  any  particular  way  of  dividing  into  sets,  say 
the  first  set  is  ab,  and  the  second  set  is  cde.  Then  keeping 
the  sets  undisturbed,  there  can  be  twelve  arrangements 
made  from  this  division  into  sets.  The  twelve  arrange- 
ments are : 

ab  cde  ba  cde 

ab  ced  ba  ced 

ab  dee  ba  dee 

ab  dec  ba  dec 

ab  ecd  ba  ecd 

ab  edc  ba  edc 


ARRANGEMENTS  AND  GROUPS.  359 

From  any  other  way  of  dividing  into  vSets  there  could 
be  twelve  arrangements  found,  hence  the  whole  number 
of  arrangements  of  five  letters  equals  twelve  times  the 
number  of  ways  of  dividing  into  sets,  or  the  number  of 
sets  equals  one-twelfth  the  number  of  arrangements. 
The  number  of  arrangements  in  this  case  is  15,  hence  the 
number  of  ways  of  making  up  sets  in  this  case  equals 
tV  15=10. 

We  will  now  take  the  general  case  of  n  letters  a,  b,  c, 
d,  <?,  /,  .  .  .  and  take  the  first  r  letters  to  form  the  first 
set,  the  following  s  letters  to  form  the  second  set,  the 
next  following  /  letters  for  the  third  set,  and  so  on. 

Place  the  letters  of  the  first  set  down  in  a  horizontal 
line,  then  those  of  the  second  set  in  the  same  horizontal 
line  following  the  first  set,  and  those  of  the  third  set  in 
the  same  horizontal  line  following  those  of  the  second  set, 
and  so  on. 

We  thus  have  all  the  n  letters  arranged  in  a  horizontal 
line,  and  it  is  evident  that  we  can  keep  these  sets  undis- 
turbed, but  still  make  several  arrangements  of  the  n  letters 
in  a  horizontal  line.  The  letters  in  the  first  set  can  be 
arranged  in  Vr  ways,  those  of  the  second  set  in  1^  ways, 
those  of  the  third  set  in  1/  ways,  and  so  on,  and  as  any 
arrangement  in  any  set  may  accompany  any  arrangement 
in  any  other  set,  hence  the  whole  number  of  arrangements 
while  the  sets  are  undisturbed  is  equal  to  Ir  U  1/  .  .  . 

Thus  from  one  way  of  making  up  the  sets  there  are 
1^- 1  ^  I  /  .  .  .  arrangements,  and  of  course  from  any  other 
division  into  sets,  there  could  be  formed!the  same  number 
of  arrangements,  hence  the  whole  number  of  arrangements 
of  n  things,  all  at  a  time,  equals  kk  k  •  •  •  times  the 
number  of  ways  of  making  up  the  sets,  or  the  number  of 
ways  of  making   up   the   sets,  equals   the  number    of 


360  UNIVERSITY   ALGEBRA. 

arrangements  divided  by  |r  [j'  [£  .  .  ,  or  the  number  ol 
ways  of  making  up  n  things  into  sets,  of  which  the  first 
contains  r  things,  the  second  s  things,  the  third  /  things, 
and  so  on,  equals 

521.  Problem.  Given  a  set  of  K  things,  another  set 
L  things,  another  of  M  things,  and  so  on;  to  find  the  num- 
ber of  groups  that  can  be  made  by  taking  r  things  from,  the 
first  set,  s  things  from  the  second  set,  t  things  from  the 
third  set,  and  so  on. 

Of  K  things  taken  r  at  a  time  there  are  -| — ir=^ — groups, 
and  of  L  things  taken  ^  at  a  time  there  are  .    .-^ groups, 

and  of  i^ things  taken  /at  a  time  there  are  ^ — ^y=f — 7 groups, 

and  so  on,  and  as  any  one  of  the  groups  from  the  first 
set  may  be  taken  with  any  one  of  the  groups  from  the 
second  set,  and  any  one  from  the  third  set,  and  so  on,  to 
form  a  larger  group,  it  follows  that  the  total  number  of 
these  larger  groups  equals  the  product 

^  [Z  ^ 

\r\K-r    \s\L-s     \t\M-t 


522.  There  are  various  relations  connecting  arrange- 
ments with  arrangements,  groups  with  groups,  groups 
with  arrangements,  etc.  We  shall  obtain  a  few  of  these 
relations,  and  we  recommend  that  the  student  try  to 
obtain  others  not  here  given. 

One  relation  was  obtained  in  Art.  510,  where  it  was 

shown  that 

^Q=;^^C-l),  (1) 


ARRANGEMENTS  AND  GROUPS.  36 1 

and  another  in  Art.  512,  where  it  was  shown  that 

AC::)  =  nAOzl).  (2) 

We  have  already  found  that 

\n 

*      ^C)=r^-  (3) 

and  from  this  it  follows  that 

^("-i)=l     L_^w  (4) 

But  \n—r-^  1  equals  the  product  of  the  integer  numbers 
from  1  up  to  ;^~r+l,  and  this  product  of  course  equals 
n—r-\-\  times  the  product  of  the  integer  numbers  from  1 
lip  to  n—r,  or 

l^---r-fl  =  («-- ^+1)    1^—^, 

hence  ^(«_,)=^l^^^  (5) 


{n-r-^V) 

Comparing  this  with  the  value  of  ^C),  equation  (3),  we 
get  A{:^^{n-r+  lM(?_i).  (6) 

If  in  (6)  we  make  r=«  we  get 

^G)=^(^i),  (7) 

or  the  number  of  arrangements  of  n  things  taken  all  at  a 
time  equals  the  number  of  arrangements  of  n  things  all 
but  one  at  a  time. 

We  have  already  found  that 

and  from  this  it  follows  that 

\n-\ 
gg-0=|  Ju._^_l  (9) 


Multiplying  both  numerator  and  denominator  of  this 
last  fraction  by  n  (n—r),  remembering  that  n  k--l=k 
and  that  (n—r)  |;g--r-- 1=  \n—r,  we  get 

(n—r)\n 
^      ^     n\r   \n^r  ^     ^ 


362 

hence,  from  (8)  and  (10), 
G0)  = 


UNIVERSITY   ALGEBRA. 


n 


From  (8)  it  easily  follows  that 


cpg:->). 


G{'r-&- 


-r+l 


(11) 


(12) 


Multiplying  both  numerator  and  denominator  of  this  last 
fraction  by  r  and  remembering  that  r  |r--l=  \r  and  that 
\n—r+l=(n—r+l)  \n'—r,  we  get 

r\n 


<^(?-l)  =  7 


(«-r+l)|r 
Comparing  (13)  and  (8)  we  easily  get 


From  (8)  it  easily  follows  thBt 
G(X.X)= 


n—r 


\n-\ 


From  (15)  and  (9)  we  get 


r-\ 


n—r 


(13) 


(14) 


(15) 


{n-r)   \n-\      r 


F 


n—r—l 


+ 


\n-l 


r— 1    \n--r 


\n-l 


n-1 


l^ 


Tr      \n—r  \r    n—r      \r   n—r     I  r  I «— r 


which  by  Art.  (514)  equals  6^(?),  hence 

c;(;)=^(r^)+6^(?ri).  (16) 

We  have  obtained  a  few  relations  connecting  arrange- 
ments with  arrangements  in  equations  (1),  (2),  (6),  (7), 
also  a  few  relations  connecting  groups  with  groups  in 
equations  (11),  (14),  (16).  We  now  obtain  a  few  relations 
involving  both  arrangements  and  groups  in  the  same 
equation. 

We  have  already  found  in  Art.  514 

^(;)=[rG(;),  (17) 


ARRANGEMENTS  AND  GROUPS.  363 

and  as  \r==A(^  we  may  write  (17)  in  the  form 

A(^}=AO  GO.  (18) 

From  (7),  -^(0=^(;:_i)  and  writing  this  value  in  (18) 
we  get  AO^^ACr.,)  GO).  (19) 

In  (18)  substitute  the  value  of  G(r)  given  in  (16)  and 
we  get   ■         A0)=AO  lGCr-')-i-GOzl)l  (20) 

But  it  readily  follows  from  (18)  that 

A0-')=AO  G(r'). 
Substituting  in  (20),  we  get 

AO')=A(r')+AC;)  GC^zl).  (21) 

Since  by  Art.  518,  groupsin  which  repetitions  are  allowed 
can  be  expressed  in  terms  of  groups  in  which  repetitions 
are  not  allowed,  it  would  be  an  easy  matter  to  obtain  equa- 
tions involving  groups  with  repetitions,  but  enough  has 
already  been  given  to  show  that  a  great  variety  of 
relations  can  be  obtained. 

KXAMPI.KS   AND   PROBIyKMS. 

1.  In  how  many  ways  can  seven  people  sit  at  a  round 
table? 

2.  How  many  different  groups  of  13  each  can  be  made 
of  52  men? 

3.  How  many  different  groups  of  two  each  can   be 
made  with  the  letters  a,  d,  I,  n,  si 

4.  How  many  arrangements  of  five  each  can  be  made 
with  the  letters  of  the  wor^  group s'>. 

5.  How  many  different  products  of  three  each  can  be 
made  with  the  four  letters  a,  b,  c,  d'- 

6.  There   are   5  straight  lines  in  a  plane,   no  two 
of  which  are  parallel;  how  many  intersections  are  there? 

7.  In  how  many  different  ways  can  the  letters  of  the 
word  algebra  be  written,  using  all  the  letters  ? 


364  UNIVERSITY   ALGFBRA. 

8.  How  many  different  signals  can  be  made  with  five 
flags  of  different  colors  hoisted  one  above  another  all  at 
a  time? 

9.  How  many  different  signals  can  be  made  with  seven 
flags  of  different  colors  hoisted  one  above  another,  five  aj; 
a  time? 

10.  How  many  different  arrangements  can  be  made  of 
nine  ball  players,  supposing  only  two  of  them  can  catch 
and  one  pitch? 

11.  How  many  different  signals  can  be  made  with  five 
flags  of  different  colors,  which  can  be  hoisted  any  number 
at  a  time  one  above  another? 

12.  In  how  many  ways  can  a  child  be  named,  supposing 
that  there  are  400  different  Christian  names,  without 
giving  it  more  than  three  names? 

13.  There  are  n  points  in  a  plane  no  three  of  which  are 
in  the  same  straight  line.  Find  the  number  of  straight 
lines  which  result  from  joining  :them. 

14.  In  how  many  ways  can  a  committee  of  3  be 
appointed  from  5  Germans,  3  Frenchmen;  and  7  Ameri- 
cans, so  that  each  nationality  is  represented? 

15.  How  many  different  signals  can  be  made  with  seven 
flags  of  which  2  are  red,  1  white,  3  blue,  1  yellow  when 
all  are  displayed  together,  one  above  another,  for  each 
signal? 

16.  On  a  railway  there  are  20  stations  of  a  certain 
class.  Find  the  number  of  different  kinds  of  tickets 
required  in  order  that  tickets  may  be  sold  at  each  station 
for  each  of  the  others. 

17.  Find  the  number  of  signals  that  can  be  made  with 
four  lights  of  different  colors,  which  may  be  displayed 
any  number  at  a  time,  arranged  either  one  above  another, 
side  by  side,  or  diagonally 


ARRANGEMENTS  AND  GROUPS.  365 

18.  There  are  n  points  in  a  plane,  no  three  of  which  are 
in  the  same  straight  line  except  r,  which  are  all  in  the 
same  straight  line;  find  the  number  of  straight  lines 
which  result  from  joining  them. 

19.  A  certain  lock  opens  for  some  arrangement  of  the 
numbers  0,  1,  2,  3,  4,  5,  6,  7,  8,  9,  taken  6  at  a  time, 
repetitions  allowed.  How  many  trials  must  be  made 
before  we  would  be  sure  of  opening  the  lock  ? 

20.  A  lock  contains  5  levers,  each  capable  of  being 
placed  in  10  distinct  positions.  At  a  certain  arrangement 
of  the  levers  the  lock  is  open.  How  many  locks  of  this 
kind  can  be  made  so  that  no  two  shall  have  the  same  key? 

21.  There  are  n  points  in  space,  no  four  of  which  are  in 
the  same  plane  with  the  exception  of  r,  which  are  all  in 
the  same  plane,  and  no  three  of  which  are  in  the  same 
straight  line.  How  many  planes  are  there,  each  con- 
taining three  of  the  points  ? 

22.  From  a  company  of  90  men,  20  are  detached  for 
mounting  guard  each  day ;  how  long  will  it  be  before  the 
same  twenty  men  are  on  guard  together,  supposing  the 
men  to  be  changed  as  much  as  possible?  How  often  will 
each  man  have  been  on  guard  during  this  time  ? 


CHAPTER  XXII. 


BINOMIAIy   THKOREJM. 


523.  The  Binomial  Theorem  enables  us  to  write 
out  any  power  of  a  binomial  without  actually  performing 
the  multiplication.  It  is*  a  deduction  from  the  gener- 
alized distributive  law  of  Art.  86. 

The  product  of  any  number  of  parentheses  is  the  aggre- 
gate of  AiyL  the  possible  partial  products  which  can  be  made 
by  taking  one  term  and  only  one  from  each  of  the  paren- 
theses.     Thus : 

ace + acf+  ade + adf-{-  bee  +  bcf-^  bde + bdf. 

524.  Binomial  Formula.  We  are  required  to  write 
out  the  value  of  {x-^ay,  where  x  and  a  stand  for  any 
two  numbers  whatever  and  ;z  is  a  positive  whole  number. 
That  is,  we  must  consider  the  product  of  the  n  parentheses 

{x-\-a^{x+a^{x-\-d)  .  .  .  (^+a). 

First.  We  may  take  an  x  from  each  of  the  parentheses 
to  form  one  of  the  partial  products.  This  gives  the 
term  x'\ 

Second.  We  may  take  an  a  from  the  first  parenthesis 
with  an  x  from  each  of  the  other  (n—V)  parentheses. 
This  gives  ax''~'^  as  another  partial  product.  But  if  we 
take  a  from  the  second  parenthesis  and  an  x  from  each  of 
the  other  {n—X)  parentheses  we  have  ax*""^  as  another 
partial  product.  I^ikewise,  by  taking  a  from  any  of  the 
parentheses  and  an  x  from  each  of  the  other  {n—V) 
parentheses  we  shall  obtain  ax*'~'^  as  a  partial  product. 
Whence,  the  product  contains  n  terms  like  ao(f'~'^,  or 
7iax''~^  is  a  part  of  the  product. 


BINOMIAL   THEOREM.  367 

ITitrd.  We  may  obtain  a  partial  product  like  a'^x''~'^ 
by  taking  an  a  from  any  two  of  the  parentheses,  together 
with  the  x's  from  each  of  the  other  (;^--2)  parentheses. 
Whence,  there  are  as  many  partial  products  like  a'^x''^'^ 
as  there  are  ways  of  selecting  2  a's  from  n  parentheses; 
that  is,  in  as  many  ways  as  there  are  groups  of  n  things 
taken  two  at  a  time,  oj 

n{n—l) 
1.2"* 

Whence,  -^ — ^  <22^«-2  jg  another  part  of  the  product. 

Fourth.  We  may  obtain  a  partial  product  like  a^xf*~^ 
by  taking  an  a  from  ^;zj/  three  of  the  parentheses  together 
with  the  x's  from  each  of  the  other  (n—Z)  parentheses. 
Whence,  there  are  as  many  partial  products  like  a^x*'~^ 
as  there  are  ways  of  selecting  3  a's  from  n  parentheses; 
that  is,  in  as  many  ways  as  there  are  groups  of  n  things 
n{n—l'){n--2') 


taken  three  at  a  time,  or 


1-2.3 


fi(fi Y)(n 2") 

Whence,  — z — ^^^—^ — -^^^""^^   is  another  part    of   the 

product. 

In  general.     We    may  obtain  a  partial   product   like 

a*^af~^  (where  r  is  a  whole  number  less  than  n)  by  taking 

an  a  from  any  r  of  the  parentheses  together  with  the  jt's 

from   each  of  the   other  (n—r)  parentheses.     Whence, 

there  are  as  many  partial  products  like  a''x''~^  as  there  are 

ways  of  selecting  r  a's  from  n  parentheses  ;  that  is,  in  as 

many  ways  as  there  are  groups  of  n  things  taken  r  at 

\n                                 \JL  _  ' 

a  time,  or  -. — r= Whence, , — j a^x"*  ''  stands  for 

*         \r  In—r  \  r\  n—r 

any  term  in  general  in  the  product  {x-\-dy*. 


368  UNIVERSITY   ALGEBRA. 

Finally,  we  may  obtain  one  partial  product  like  a*"  by- 
taking  an  a  from  each  of  tbe  parentheses. 
Whence,  «"  is  the  last  term  in  the  product. 
Thus  we  have  proved 

or  {x+a)'*==x*'+naa>**-^+'^^!^^!^^  .. 

\n 

The  expression  on  the  right-hand  side  of  the  equation 
is  called  the  Expansion  or  Development  of  the  power 
of  the  binomial. 

525.  The  expansion  of  (xdtay  is  usually  called  the 
Binomial  Formula. 

If  in  the  result  of  the  last  Article,  we  substitute  dta 
for  a,  we  get  the  following  as  the  expansion  of  (xdha)**: 

2  2.3 

^n(n-1)(n-2)(n-3)^4^„4^    _  ^j 

Therefore,  in  any  power  of  the  difference  of  two  num- 
bers the  sign  of  the  first  term  is  +,  of  the  second  — ,  and 
so  on,  alternately  +  and  — . 

526.  Binomial  Theorem.  By  observing  carefully 
the  expansion  of  (x+a)**  written  above  it  will  be  seen 
that  we  may  state  the  binomial  formula  in  the  form  of  a 
theorem  as  follows : 

I.  Exponents.  In  any  power  of  a  binomial,  x+a,  the 
exponent  of  x  commences  in  the  first  term  with  the  expojient 
of  the  required  power,  and  in  the  following  terms  contin- 
ually decreases  by  unity.  The  exponent  of  a  commences  with 
1  in  the  second  term  of  ^he  power,  and  conthiually  increases 
by  unity. 


BINOMIAL    THEOREM.  369 

II.  COKFFICIKNTS.  The  coefficient  in  the  first  term  is  1, 
that  in  the  second  term  is  the  exponent  of  the  power ;  and 
if  the  coefficient  in  any  term  be  multiplied  by  the  exponent 
of  X  in  that  term  and  divided  by  the  exponent  of  a  increased 
by  ly  it  will  give  the  coefficient  in  the  succeeding  term,. 

527.  Below  we  give  a  few  examples  of  the  use  of  the 
binomial  theorem. 

(1)  Expand  («+<5)«. 

We  may  expand  this  at  once  by  the  theorem  as  follows : 

or  we  may  substitute  x=^a,   a=b,    and   n~^   in   the   formula    [1], 
obtaining  : 

+2.3.4.5         ^2.3.4.5.6 
which  reduces  to  the  same  result  as  above. 

(2)  Expand  {u  +  Sy)^. 

Here  x=u  and  a=3y.     By  the  theorem  we  get 

«5  4-5^^4(3/)  +  102^3(3^)2_|_  10^^2(3y)3  4.5^^(3^)4  _|_  (3^)5^ 

Performing  the  indicated  operations,  we  get 

u^  +  15uy^90u^y^+210u^y^+i05uy*+2idy^, 

(3)  Expand  (^2 -2)*.  ^-«T    lo    'r^dmuM      MSM 
Here  x=r^,  a=—2,  and  «=4.     By  the  theorem 

(r2)4_4(;.2)82+6(r2)222-4(r2)23+24. 
Performing  the  indicated  operations,  we  get 

rs  -  8^6  +  24^4  -  32^2  + 16. 

(4)  Expand  (3^-i)8. 

HeTex=db,  az=—^,  n=S.     By  the  theorem    ... 
(3^)3-3(3^)2(i)  +  3(3^)(4)2-(4)8r 
Performing  the  indicated  operations,  we  get 

24 -U.  A. 


370  UNIVERSITY    ALGEBRA. 

KXAMPLKS. 

Expand  each  of  the  following  by  the  binomial  theorem 
or  formula : 

1.  (a+xy.  9.   (1  +  ^)'^  17.   (m-hky. 

2.  (d+xy,  10.   (2+xy.  18.   (a-xy. 

3.  (d+yy.  II.   (2-xy,  19.   (ddSyy, 

4.  (C+Xy,  12.    (i  +  ^)^  20.    (^3_^2)8^ 

5.  (-X+2ay,         13.     (^2_^2)5^  21^     (2/^2_3^3)5^ 

6.  (2;t;+3a)«.        14.  (:r+2^)^  22.   (Sx^-iy, 

7.  (1-.^)^  15.   (Sa+^y.  23.   (i/^4-^2'- 

8.  (1--^)^  16.  (2ax-x^y       24.   (3^-l/;r)^ 

2        3 

25.  (;t:_y— ^^)^.  30.   (.r'5'+;»;2")6. 

26.  (l/^-f/^)6.  31.    («-2-3^4. 

27.  (ia+lx+y]y.  32.      (^2  4.2^;|;+^2)8^ 

28.  ([a+^]-2)».         33.  {v^^^^y. 

29.  (a+^-J/)^  34.  {^+^2}  • 

PROPERTIES  OF  THE  EXPANSION. 

528.  Number    of   Terms.      The  exponents  of  a 

through  the  binomial  formula  constitute  the  following 

scale : 

0,  1,  2,  3,  4,  ...  ^2. 

The  number  of  terms  in  this  scale  is  n+1.     Therefore 

the  number  of  terms  in  the  expansion  of  (^x+aY  is  n+1. 

529.  Value  of  the  r  th  Term.     The  general  term  in 
the  expansion  of  (x+ay  we  have  seen  to  be 

[n 
I  r\  n—r 


BINOMIAL   THEOREM.  3/1 

By  noting  the  exponent  of  a,  this  is  seen  to  be  the 

r-\- 1  term  from  the  beginning,  since  Xh.^  first  term  contains 

the  zero  power  of  a.     Therefore  the  r  th  term  in  the 

expansion   will    be   found   from   the   general    term    by 

decreasing  the  r  in  it  by  unity.     This  gives 

\n 

r  th  term  in  (a; + »)"=  i == «;"-''+  ^ a"" ^  [2] 

I  r— 1     [w.-r+l 

530.  Coefficients  Equidistant  from  the  Ends.   We 

have  just  found  that  the  coefficient  of  the  r  th  term  from 
the  beginning  in  the  expansion  of  {x-\-ay  is 
|_^ 
I  r—\  1 7^— r+1 


Since  there  are  n-\-\  terms  altogether,  there  must  be 
(72  +  1)— r  terms  before  the  r  th  term  from  the  end;  that 
is,  the  r  th  term  from  the  end  is  the  n—r-\-^  term  count- 
ing from  the  beginning.  Substituting  n—r-\-^  for  r  in 
the  expression  for  the  r  th  term,  we  get  the  following  value 
fro  the  r  th  term  from  the  end : 
\n 

— - ^r—  1  >y«— r-f- 1 

\n^r^-\\r--r'      "" 
Thus  we  note  that  in  the  expansion  of  (x+a)**  the 
coefficients  of  terms  equidistant  from  the  ends  are  equal, 

531.  Coefficients  of  Odd  and  Even  Terms.  In 
the  expansion  of  {x-^rcCy  put  x=^  1  and  a=  —  1.  We  thus 
obtain 

ft_^   ,  <^-l)  ,  n{n^V){n^^Xn-^^  ,  \ 

V  "^    1  .  2    "^         1.2.3.4         •+"•••; 

which  shows  that  the  sum  of  the  coefficients  of  the  first, 
third,  fifth,  .  .  .  terms  equals  the  sum  of  the  coefficients 
of  the  second,  fourth,  sixth,  .  .  .  terms. 


372  UNIVERSITY   ALGEBRA. 

In  the  binomial  formula  the  sum  of  the  coefficients  of  the 
even  terms  equals  the  sum  of  the  coefficients  of  the  odd  terms. 
The  above  result  may  be  also  written 

^G)+6^e2)+^a)+. .  .=^a)+6^(3)+^(5)+. . . 

where  both  sides   of  the  equality  are  to  be  continued 
until  6^(2)  occurs  on  one  side  or  the  other. 

532.  Sum  of  the  Coefficients.  In  the  expansion 
of  {x+dy  put  x=-\  and  «=1.     We  then  have 

That  is,  the  sum  of  all  the  coefficients  in  the  expansion  of 
(x-\-ay  equals  2". 

533.  MidVile  Term.  If  the  expansion  consists  of 
an  odd  number  of  terms,  there  is  a  middle  term.  That 
is,  there  is  a  middle  term  if  n-\-\  is  odd;  that  is  if  ^  is 
even.  Since  the  coefi&cients  of  terms  equidistant  from 
the  ends  are  equal,  we  may  write 

2'*=2(l+;z+^^^-+.  •  .  to  I  terms)  +  middle  term. 

In  order  that  this  equation  may  be  true,  the  middle 
term  must  be  divisible  by  2.  That  is,  the  coefficient  of  the 
middle  term  in  the  binomial  formula  is  divisible  by  2. 

KXAMPl^KS. 
I.  Find  the  n  th  term  of  {nT—n"**)**, 


f  1    \i* 

2.  Find  the  11th  term  of    4;i; --A     ' 

V         2Vx/ 


3.  Write  down  the  10th  term  of  (a—xy^. 

4.  Write  down  the  middle  term  of  (1+^)^". 

5.  Write  down  the  term  containing  .;*:'  in  the  expansion 
of(l+xy. 


BINOMIAL   THEOREM.  373 

6.  Write  down  the  two  middle  terms  in  the  expansion 

7.  Write  down  the  two  middle  terms  in  the  expansion 


»K'+^) 


2M-1 


MUI.TINOMIAI.   THEORKM. 


534.  The  expansion  of  any  power  of  a  polynomial, 
a-{-b-\-c+  .  .  .  can  be  found  by  successive  applications 
of  the  binomial  formula.  Thus,  (^a-{-b+c+d+  .  ,  ,y  is 
the  same  as  (a+ld+c-{-d+  .  .  .])'*.  The  general  term  in 
the  expansion  of  this  latter  is  by  Art.  529. 

In  the  same  way  the  general  term  in  the  expansion  of 
{l>+c+d+.  .  .y-  or  ib+[c+d+.  .  .])«-"  is 

^-l>'(c-i-d+.  .  .  )«-'-^  ^2) 


\L 


Similarly,    the    general    term    in    the    expansion    of 
ic+d+.  .  .y-"--'  or  (r+[^+.  .  .])'*— ^  is 
\n — r — s 
,     '1  /(^+.  .  .)""""""',  (3) 

and  so  on.     Whence,  the  general  term  in  the  expansion 
of  {a-^-b+c-^-d^-,  .  .y  is  the  product  of  (1),  (2),  (3),  etc., 

\n  \n — r  I  n — r — s 


or, 


-X 


X 


Vr  n—r     \s     n—r—s     T7 


n—r—s—f 


.  oTb'c^. 


which,  simplified,  gives 

It  is  evident  from  the  process  of  the  formation  of  [3] 
that  the  numbers  r,  s,  t,  .  .  .  may  stand  for  any  positive 
whole  numbers  whatever,  provided  that 
r-\-s+t+  .  .  .  =«. 


374  UNIVERSITY   ALGEBRA. 

535.  General  term  in  (a  +  dx-\-cx'^+dx^+  .  .  .y\ 
This  can  be  written  down  at  once  from  the  result  just 
obtained.     The  general  term  is  evidently 

t 


\g\r    ^    \l.. 


-a^bxYicx'^yidx^y. 


\n 
that  is,     -. — I — ri=-i-: a^b-c'd'.  .  .  x'+^'+^'+  '  '  • 

if  [r  If  li--- 

In  this,  as  before, 

q+r+s+i+,  .  .=;^. 

EXAMPI^KS. 

1.  Write  out  the  coefficient  of  x^  in  the  expansion  of 

(2+X'-x''y. 

The  general  term  is 

., \-  ,    2^(-lK^>-+2J, 

\i  Lrii- 

Now,  g-{-r-{-s=5,  (1) 

and  r+2j=8.  (2) 

Whence,  we  may  have  from  (2)  r=0,  s=4;^r=2,  J=3;  r=:4,  ^=2; 
r=Q,  s=l',  r— 8,  ^=0.  Of  these,  however,  only  r=0,  J=4;  ^=2,  ^=3 
will  go  with  (1).     Whence  we  say 

r=0,  J3=4,  ^=1, 
and  r=2,  ^=3,  ^=0. 

which  gives  the  terms 

L5  li 

The  sum  of  these  two  terms  is  0.  Whence,  the  coefi&cient  of  x^ 
in  the  expansion  of  (2  +  ^— :«;*)** is  0 

2.  Write  out  the  coefficient  of  a^d^c^  in  the  expansion 
of  (ia-\-d-\-cy^, 

3.  Find   the   coefficient  of   x^   in   the  expansion  of 

4.  Find  the  coefficient  of   x^    in   the   expansion  of 


BINOMIAL   THEOREM.  375 

5.  Show  that  the  coefficient  of  x^  in  the  expansion  of 
(l  +  2x+x^y  is  56. 

6.  Write  out  the  coefficients  of  a^b^,  and  d^cd  in  the 
expanvSion  of  (a+d+c+d)^. 

Historical  Note.  The  Binomial  Formula  is  engraved  upon 
the  tomb  of  Sir  Isaac  Newton.  (1640-1727)  in  Westminster  Abbey. 
He  discovered  it  while  he  was  a  student  at  Cambridge.  It  grew  out 
of  the  study  of  Wallis's  investigations  on  the  quadrature  of  curves. 
Newton  gave  no  formal  proof  of  the  theorem.  Indeed,  not  until  the 
present  century  were  rigorous  proofs  discovered,  applicable  to  the 
case  when  the  expansion  becomes  an  infinite  series. 

Near  approaches  to  the  discovery  of  the  Binomial  Formula  for  the 
case  of  positive  integral  exponents  were  made  before  the  time  of 
Newton.  Vieta  (1540-1603)  gave  the  rule  for  obtaining  the  powers  of 
a  binomial.  He  observed  as  a  necessary  result  of  the  process  of  mul- 
tiplication that  the  successive  coefficients  of  any  power  of  a  binomial 
are:  first,  unity;  second,  the  sum  of  the  first  and  second  coefl&cients 
in  the  preceding  power;  third,  the  sum  of  the  second  and  third  coeffi- 
cients in  the  preceding  power,  and  so  on.  Vieta  noticed  also  the 
uniformity  in  the  product  of  binomial  factors  of  the  form  of  x  +  a, 
x-\-d,  x-\-c,  etc.  But  Harriot  (1560-1621)  independently  and  more 
fully  treated  of  these  products  in  showing  the  nature  of  the  composi- 
tion of  a  rational  integral  equation.  In  this  connection  it  is  interesting 
to  note  that  Harriot  was  the  first  mathematician  to  transpose  all  the 
terms  of  an  equation  to  the  left  member. 


CHAPTER  XXIII. 

THEORY   OF  PROBABII^ITIKS. 

636.  Mathematical  Probability.  There  is  a  large 
class  of  events  of  which  our  knowledge  is  insufficient  to 
permit  us  to  predict,  in  a  specified  case,  the  precise  thing 
that  will  happen,  yet  of  which  our  knowledge  is  sufficient 
to  enable  us  to  express,  in  a  quantitative  way,  what  ought 
to  be  our  belief  concerning  the  happening  of  the  event. 
Thus,  if  a  coin  be  tossed,  our  knowledge  is  not  sufficient 
to  enable  us  to  say  whether  it  will  fall  ' '  heads ' '  or 
whether  it  will  fall '*tails*',  but  our  knowledge  is  sufficient 
to  enable  us  to  say  that  //  zs  just  as  likely  to  fall  heads  as 
tails.  Likewise,  there  is  another  class  of  events  in  which 
we  cannot  make  a  definite  statement  concerning  any  one 
event  of  a  group,  yet  concerning  the  group  itself  we  may 
be  able  to  predict  this  or  that  fact  with  considerable 
approximation  to  accuracy.  For  example,  nothing  can 
be  more  uncertain  than  to  say  that  a  certain  man  aged 
30  and  now  in  good  health  will  be  alive  10  or  20  years 
hence.  Yet  of  a  large  number  of  such  men  we  may  assert, 
with  safety,  that  84  per  cent,  will  be  alive  10  years  hence. 
If  the  number  be  10,000  the  assertion  may  be  made  with 
greater  confidence  than  if  the  number  be  1,000;  thus, 
though  each  individual  case  is  most  uncertain  in  its  out- 
come, yet  a  very  large  group  of  these  uncertain  cases 
may  be  treated  with  almost  as  much  confidence  as  if 
dealing  with  the  law  of  gravity  itself. 

A  quantitative  expression  which  measures  what  our 
belief  ought  to  be  concerning  an  event  of  either  of  the 
above  classes  in  which  absolute  knowledge  is  impossible, 


THEORY   OF  PROBABILITIES.  377 

is  called  the  mathematical  probability  of  the  event,  and 
may  be  defined  as  follows : 

If  an  event  is  in  question  in  a  +  d  cases,  all  of  which 
are  equally  likely,  and  if  in  a  of  these  cases  the  event 
will  happen,  and  in  the  remaining   cases  will    fail    to 

happen,  then  the  Probability  of  its  happening  is  — -r-v 

and  the  probability  of  its  failing  is  —rrr 

Thus,  suppose  that  an  ordinary  die  (which  has  six 
faces  numbered  from  1  to  6)  be  thrown;  what  is  the 
probability  that  a  certain  face,  say  5,  will  be  uppermost? 
Since  the  die  has  six  faces,  there  are  six  cases  in  question, 
all  of  which  are  equally  likely;  that  is,  we  have  no  reason 
to  expect  one  rather  than  another  of  the  faces  to  come  up. 
In  one  of  these  cases  the  event  (throwing  a  5)  happens 
and  in  the  other  five  cases  the  event  fails  to  happen. 
Whence  the  probability  of  throwing  a  5,  by  definition,  is  \. 

If  a  die  be  thrown,  what  is  the  probability  of  getting  an 
even  number?  As  before,  there  are  six  cases  in  question, 
all  of  which  are  equally  likely,  and  in  three  of  which  the 
event  happens.  Whence,  the  probability  of  throwing  an 
even  number  is  f  or  \. 

If  a  die  be  thrown,  what  is  the  probability  of  getting 
less  than  5?  Here  there  are  6  equally  likely  events  in 
question,  in  4  of  which  the  event  happens.  Whence, 
the  probability  is  f . 

'  •  I  prefer  to  say  that  the  theory  of  probability  deals  with  quantity 
of  knowledge.  An  event  is  only  probable  when  our  knowledge  of  it  is 
diluted  with  ignorance,  and  exact  calculation  is  needed  to  discriminate 
how  much  we  do  and  do  not  know.  It  teaches  us  to  regulate  our 
action  with  regard  to  future  events  in  a  way  which  will,  in  the  long 
run,  lead  to  the  least  amount  of  disappointment  and  injury.  It  is,  as 
Laplace  as  happily  expressed  it,  good  sense  reduced  to  calculation. ' ' 

"The  theory  consists  in  putting  similar  cases  upon  a  parj  and 
distributing  equally  imong  them  whatever  knowledge  we  may  possess. 


378  UNIVERSITY   ALGEBRA. 

Throw  a  penny  into  the  air,  and  consider  what  we  know  with  regard 
to  its  mode  of  falling.  We  know  that  it  will  certainly  fall  upon  a  flat 
side,  so  that  the  head  or  tail  will  be  uppermost,  but  as  to  whether  it 
will  be  head  or  tail  our  knowledge  is  equally  divided.  Whatever  we 
know  concerning  head,  we  know  as  much  concerning  tail,  so  that  we 
have  no  reason  for  expecting  one  more  than  the  other.  The  least 
predominance  of  belief  to  either  side  would  be  irrational,  and  would 
consist  of  treating  unequally  things  of  which  our  knowledge  is 
equal." — Quoted,  with  omissions,  from  W.  S.  Jevons'  ''Principles  of 
Science. ' ' 

537.  Broader  Definitions.  Problems  like  those  in 
insurance,  where  the  probability  must  be  estimated  by- 
means  of  data,  secured  by  observation  covering  a  large 
number  of  cases,  require  a  broadening  of  the  above  defi- 
nition. To  illustrate  the  cases  in  which  the  enumeration 
of  the  number  of  cases  in  question  is  impracticable, 
Bertrand  asks,  ''What  is  the  probability  that  the  Seine 
will  be  frozen  at  Paris  in  the  course  of  1995  ?*' 

In  this  connection  the  following  definitions  of  probability  are 
valuable:  If  on  taking  any  very  large  number  n  out  of  a  series  of 
cases  in  which  an  event  A  is  in  question,  A  happens  on  pn  occasions, 
the  probability  of   the  event  A  is  said  to  be  p. — Crystal's  Algebra. 

' '  The  probability  of  an  event  is  the  ratio  of  the  favorable  cases  to 
the  total  number  of  cases  possible;  a  condition  is  understood:  all  the 
cases  must  be  equally  likely." — Preliminary  definition,  Bertrand' s 
Calcul  des  Probabi life's.  ' '  The  probability  of  an  event,  whatever  its 
nature  may  be,  is  said  to  equal  a  given  fraction  /,  when  he  who  awaits 
the  event  might  exchange,  indifferently,  the  fears  or  hopes,  the  advan- 
tages or  disadvantages  incident  to  the  happening  of  the  event,  for  the 
consequences  (supposed  identical)  of  drawing  a  ball  from  an  urn 
whose  composition  gives  rise  to  a  probability  equal  to  /." — Bertrand, 
subsequent  definition. 

538.  Certainty.  If  an  event  happens  in  every  case 
in  which  it  is  in  question,  without  fail,  then  the  proba- 
bility of  the  event,  by  definition,  is  — ^  or  1.  Thus, 
certainty,  in  accordance  with  our  definition,  is  denoted  by 


THEORY    OF    PROBABILITIES.  3/9 

unity.  For  example,  what  is  the  probability  of  throwing 
less  than  10  with  a  common  die?  There  are  6  events  in 
question,  in  all  of  which  the  event  happens.  Therefore, 
the  probability  is  1  or  certainty. 

Also,  if  an  event  happens  in  no  case  in  which  it  is  in 
question,  then  the  probability  of  the  event  is,  by  definition, 

^r — I  or  0.     Thus,  impossibility,  in  accordance  with  our 

definition,  is  denoted  by  0. 

539.  Complementary  Events.  If  p  stands  for  the 
probability  of  an  event  happening,  then  \—p  is  the 
probability  that  the  event  fails  to  happen;  for  evidently 

-^   the  probability  of  the  event  failing  to   happen, 

equals  1-^^- 

Thus  we  note  that  the  probability  of  an  event  is 
always  expressed  by  a  positive  proper  fraction  —  a  num- 
ber between  0  and  1 . 

540.  Odds.     Instead  of  saying  that  the  probability 

of  an  event  is  — — ,  we  may  say  that  the  odds  are  a  \,o  b 
a-\-o 

hi  favor  of  the  event  or  that  the  odds  are  b  X.o  a  against 
the  event.  Thus,  instead  of  saying  that  the  probability 
of  throwing  less  than  3  with  a  single  die  is  \,  we  may  say 
that  the  odds  are  1  to  2  in  favor  of  throwing  less  than  3 
or  2  to  1  against  throwing  less  than  3. 

SIMPLE   PROBABILI'TY. 

541.  Many  problems  in  probability  involve  nothing 
but  an  application  of  the  definition  of  probability  and  a 
knowledge  of  the  theorems  in  arrangements  and  groups. 

(1)  Two  dice  are  thrown.  What  is  the  probabiUty  of  throwing 
twoC's  ?     A  5  and  a  6  ?     A  sum  greater  than  9  ?     A  sum  equal  to  10? 


38o  UNIVERSITY    ALGEBRA. 

When  two  dice  are  thrown  the  number  of  events  in  question  is  the 
number  of  ways  of  arranging  the  faces  taken  2  at  a  time,  repetitions 
allowed,  or  6^.  One  of  these  is  the  case  in  which  the  getting  of  two 
6's  occurs.     Whence  the  probability  of  throwing  two  6's  is  ^. 

A  5  and  a  6  may  occur  in  two  ways ;  the  first  die  may  turn  5  and 
the  second  6,  or  the  first  die  may  turn  6  and  the  second  5.  Whence, 
the  probability  of  a  5  and  a  6  is  3%  or  ^. 

A  sum  greater  than  9  may  be  made  in  the  following  ways : 

First  die,  6  6  5  5  6  4 

Second  die,  6  5  6  5  4  6 

Sum.  12        n        11        To        10        10 

or  6  ways  in  all.     Whence,  the  probability  of  throwing  more  than  9 

is  3^  or  ^. 

A  sum  equal  to  10  can  be  made  in  three  ways.  Whence,  the 
probability  of  throwing  exactly  10  is  f^  or  ^. 

(2)  Two  balls  are  to  be  drawn  from  a  bag  containing  4  white  and  6 
black  balls.     What  is  the  probability  that  both  will  be  white  ? 

Altogether  there  are  G{^^)  different  pairs  which  may  be  drawn,  all 
of  which  are  equally  likely.  In  G{^)  of  these  both  are  white.  Whence, 
the  required  probability  is 

4 .  3  .  10  .  9      ^ 
1.2*    1  .  2  °^  15* 

(3)  Thirty  coins  are  thoroughly  shaken  and  thrown.  What  is  the 
probability  that  all  turn  heads  ? 

Thirty  coms  can  fall  in  2^®  ways.  The  event  (all  turning  heads) 
happens  in  but  one  of  these  ways.  Whence,  the  required  probability  is 

p^=.000  000  000  93133. 

The  probability  is,  of  course,  the  same  as  getting  all  heads  in  30 
throws  with  a  single  coin. 

(4)  A  ball  is  to  be  drawn  from  an  urn  containing  4  white,  6  red, 
and  10  black  balls.     What  is  the  probability  of  either  white  or  black? 

The  probability  of  not  drawing  white  or  black  (or  the  probability  of 
drawing  red)  is  ^%  or  ^5.  Whence,  the  probability  of  drawing  white 
or  black  is  1— x^o  or  ^^. 

EXAMPI^KS. 

I.  An  urn  contains  3  white,  4  red,  and  7  blue  balls. 
If  three  be  drawn  what  is  the  probabilit}^  of  getting  red, 
white,  and  blue? 


THEORY    OF   PROBABILITIES.  38 1 

The  total  number  of  cases  in  question  is  the  number  of  arrangements 
of  14  things  three  at  a  time,  or  ^(V).  The  number  of  cases  in  which 
the  event  happens  is  the  number  of  arrangements  of  3  things  taken  all 
at  a  time  multiplied  by  3X4X7. 

2.  What  is  the  chance  of  throwing  one  6  and  only  one 
in  a  single  throw  of  two  dice  ? 

The  number  of  cases  in  question  is  36.  The  cases  in  which  the 
event  occurs  are  when  the  6  of  the  first  die  falls  with  one  of  the  other 
five  faces  of  the  second  die,  and  when  the  6  of  the  second  die  falls 
with  one  of  the  other  five  faces  of  the  first  die. 

3.  An  urn  contains  3  white,  4  red,  and  7  blue  balls. 
If  three  be  drawn  in  succession,  what  is  the  probabilitj'-  of . 
getting  the  order,  red,  white,  and  blue? 

4.  What  is  the  probability  of  throwing  doublets  with 
two  dice? 

5.  If  four  coins  be  tossed,  what  is  the  probability  that 
exactly  half  will  fall  heads  ? 

The  number  of  cases  in  question  is  2*  or  16.  The  number  of  cases 
in  which  the  event  happens  is  the  number  of  arrangements  of  two  «'s 

14 
and  two  b's  or  — = — , 

\%     ^ 

6.  When  two  dice  are  thrown,  what  is  the  probability 
of  not  throwing  an  ace  ? 

7.  A  set  of  dominoes  is  numbered  from  double  blank 
to  double  six.  If  one  be  drawn  at  random,  what  is  the 
probability  that  it  contains  a  6  ? 

The  number  of  dominoes  in  the  set  is  the  number  of  groups  of  seven 

7    8 
things  taken  2  at  a  time,  repetitions  allowed,  or  — '- — . 

2 

8.  If  four  cards  be  drawn  from  a  pack,  what  is  the 
probability  that  there  will  be  one  of  each  suit? 

9.  In  a  ba^  there  are  4  white  and  6  black  balls.  What 
is  the  probability,  if  they  be  drawn  in  succession,  that 


382  UNIVERSITY   ALGEBRA. 

the  white  balls  will  all  be  drawn  first  and  then  the  black 
balls? 

The  number  of  cases  in  question  is     ' — -. 

\±    [^ 

10.  In  a  bag  there  are  4  white  and  6  black  balls.  If  3 
be  drawn,  what  is  the  probability  that  all  will  be  black  ? 

11.  Show  that  the  probability  of  throwing  more  than 
15  in  one  throw  with  three  dice  is  less  than  ■^. 

12.  Four  cards  are  missing  from  a  pack.  What  is  the 
probability  that  there  is  one  from  each  suit  ? 

13.  Thirteen  persons  take  places  at  a  round  table. 
Show  that  it  is  5  to  1  against  two  specified  persons 
sitting  together. 

14.  What  is  the  probability,  in  throwing  two  dice  three 
times  in  succession,  of  obtaining  a  doublet  at  least  once? 

The  number  of  cases  in  question  is  36^  or  46656.  At  each  throw 
6  of  the  36  possible  cases  gives  doublets.  Whence,  the  number  of 
cases  which  give  no  doublets  in  three  throws  is  30^  or  27000.  The 
number  of  cases  which  contain  at  least  one  doublet  is  then 

363-303  =  19656. 
Whence,  the  required  probability  is 

i?555=.4213    .. 
46656 

15.  If  4  «'s  and  3  ^'s  be  placed  in  a  row  at  random, 
show  that  the  probability  of  the  first  and  last  letters  being 
both  ^'s  is  \, 

TOTAI.  PROBABII^ITY. 

542.  Probability  has  been  defined  as  the  ratio  of  the 
number  of  cases  in  which  an  event  happens  to  the  total 
number  of  cases  in  question.  If  the  cases  in  which  the 
event  happens  are  divided  into  several  classes,  the  proba- 
bility of  the  event  will  be  the  sum  of  the  probabilities 
pertaining  to  each  of  the  classes. 


THEORY   OF   PROBABILITIES.  383 

Thus,  the  probability  of  throwing  more  than  14  with 
three  dice  is  the  sum  of  the  probabilities  of  throwing  15, 
16,  17,  and  18.  Likewise,  if  A's  probability  of  taking  a 
prize  is  \,  and  B's  probability  of  taking  it  is  -|-,  and  C's 
probability  of  taking  it  is  |,  then  the  probability  that 
either  A,  B,  or  C  takes  the  prize  is  i+-|-+|  or  ^.  The 
probability  of  some  one  else  taking  it  is  then  ^. 

543.  The  division  into  classes,  as  supposed  above,  is 
arbitrary,  with  the  implied  condition  that  the  division 
must  be  made  so  as  to  include  all  of  the  cases  in  which 
the  event  happens,  without  including  any  case  more 
than  once.  To  express  this  condition,  we  say  that  all 
the  events  belonging  to  the  different  classes  must  be 
Mutually  Exclusive;  that  is,  the  supposition  that  any 
one  of  the  events  happens,  must  be  incompatible  with  the 
supposition  that  any  other  happens. 

Thus,  the  probability  of  throwing  a  3  or  a  4  with  two  dice  is  noi 
the  sum  of  the  probabilities  of  throwing  a  3  and  a  4.  The  number  of 
•events  in  question  is  36.  The  number  of  cases  in  which  a  3  is  thrown 
is  11  and  the  same  for  a  4.  But  in  estimating  the  cases  in  which  a  3 
occurg  we  counted  the  throws  3,  4  and  4,  3.  Evidently  these  should 
not  be  recounted  in  estimating  the  number  of  cases  in  which  a  3  or  a 
4  is  thrown. 

Containing  3,  *  *         xxxxxx       *  ^^  ^^ 

-r^.^xxro  J  1st  die,  mill  222222  333333  444444  555555  666666 
inrows,  -j  2nd  die,  123456  123456  123456  123456  123456  123456 
•Containing  4  *  *  *        xxxxxx         *  ^^ 

The  required  probability  is  |J  or  |  and  not  |J,  The  events  sup- 
posed are  not  mutually  exclusive,  since  the  supposition  that  a  3  turns 
is  not  incompatible  with  the  supposition  that  a  4  turns,  for  they  may 
both  turn  at  once. 

KXA.MPI,ES. 

I.  If  the  probability  that  a  shot  aimed  at  a  target  hits 
the  bull's  eye  be  ^,  the  probability  that  it  hits  the  first 
ring  be  |,  and  the  probability  that  it  hits  the  outer  ring 
he  \,  what  is  the  probability  that  it  hits  the  target  at  all? 


384  UNIVERSITY   ALGEBRA. 

The  rivents  are  mutually  exclusive  and  the  event  of  hitting  the 
target  can  be  decomposed  into  these  three  groups  of  events.  There- 
fore, the  required  probability  is  1*5+ J+ J  or  J. 

2.  If  the  probabilty  of  A  winning  a  race  be  -|-,  and  the 
probability  of  B  winning  be  ^^  what  is  the  probability 
that  either  A  or  B  wins? 

3.  A  bag  contains  4  red,  8  black,  12  white,  and  16  blue 
balls.  What  is  the  probability,  if  one  be  drawn,  of  get- 
ting either  red,  white,  or  blue  ? 

4.  If  a  ticket  be  drawn  from  a  set  numbered  from  1  to 
30,  what  it  its  probability  that  its  number  will  not  be  a 
multiple  of  6  or  of  7  ? 

5.  When  two  dice  are  thrown,  what  is  the  probability 
that  the  throw  will  be  greater  than  8  ? 

6.  If  the  probability  of  A  taking  a  prize  be  ^,  and  the 
probability  of  B  taking  it  is  I-,  what  is  the  probability 
that  neither  takes  it? 

COMPOUND  PROBABIIvlTY. 

544.  Several  single  events  occurring  simultaneously 
or  in  succession  may  be  considered  as  constitutifig  a 
Compound  Event.  Thus,  the  drawing  of  a  red,  a 
white  and  a  blue  ball  in  succession  may  be  considered  a 
compound  event  composed  of  the  three  simple  events 
mentioned. 

Events  are  said  to  be  Dependent  or  Independent 
according  as  the  occurrence  of  one  does  or  does  not  affect 
the  occurrence  of  the  others. 

545.  The  probability  that  two  independent  events  will 
both  happen  is  the  product  of  their  respective  probabilities 
of  happening. 

Let  a  be  the  number  of  ways  in  which  the  first  event 
may  happen,  and  b  the  number  of  ways  in  which  it  may 


THEORY   OF   PROBABILITIES. 


385 


fail,  all  these  ways  being  equally  likely  to  occur.  Also, 
let  a^  be  the  number  of  ways  in  which  the  second  event 
may  happen,  and  dj^  the  number  of  ways  in  which  it  may 
fail,  all  these  ways  being  equally  likely  to  occur.  Each 
case  out  of  the  a  +  d  cases  may  be  associated  with  each  case 
out  of  the  ^1  +  ^1  cases.  Thus,  there  are  (a  +  ^)(«i+<^i) 
compound  cases  which  are  equally  likely  to  occur. 

In  auj^  of  these  compound  cases  both  events  happen, 
in  ddj^  of  them  both  events  fail,  in  ad^  of  them  the  first 
event  happens  and  the  second  fails,  and  a^^d  of  them  the 
first  event  fails  and  the  second  happens.     Thus,  we  have 


THE  FRACTIONS. 

THK  RKSPKCTIVB  PROBABII^lTi:^ 

^^1 

That  both  events  happen. 

That  both  events  fail. 

That  first  event  happens  and  second  fails. 

That  first  event  fails  and  second  happens 

ab^ 

{a-^b){a^-^b^) 
a-^b 

(a  +  ^)(^,  +  ^) 

If  p  and  />!  stand  for  the  probabilities  of  the  happening 
of  the  two  events,  we  have  from  above  : 

The  probability  that  both  events  happen  equals  pp-^^. 
The   probability   that  both  of  the  events  fail  equals 

(i-/)(i-/0. 

The  probability  that  first  event  happens  and  second 
fails  equals /(I —/>i). 

The  probability  that  first  event  fails  and  second  hap- 
pens equals  iX~P)P\' 

25  — u.  A. 


386  UNIVERSITY    ALGEBRA. 

(1)  The  probability  that  A  can  solve  a  certain  problem  is  J,  and 
that  B  can  |solve  it  is  |.  If  they  both  try,  find  the  probability 
(1)  that  they  both  succeed ;  (2)  that  A  succeeds  and  B  fails ;  (3)  that 
A  fails  and  B  succeeds  ;  (4)  that  both  fail ;  (5)  that  the  problem  will 
be  solved. 

Here  we  have  p  =i  ^—p  =f- 

Probability  both  succeed  =:pp^-=^. 

Probability  A  succeeds,  B  fails  =/{!  -/i)=i^. 
Probability  A  fails,  B  succeeds  ={\~p)p^=:^. 
Probability  both  fail  =(1  -/)(!  -p^  )=\. 

i+i^  +  i  +  i=H==l  as  it  should, 
The  probability  that  the  problem  is  solved  is  the  probability  that 
one  at  least  succeeds  in  solving  it,  or  1  —  (1— /)(!— /i)=|. 

(2)  A  man  draws  a  ball  from  each  of  two  urns.  The. first  urn  con- 
tains 4  white  and  7  black  balls  and  the  second  contains  7  white  and  3 
red  balls.     What  is  the  probability  of  drawing  two  white  balls  ? 

The  events  in  question  are  independent  and  their  respective  proba- 
bilities are  ^  and  /g.  Whence,  the  probability  of  drawing  two 
whiteisi^X/oorif. 

(3)  In  the  same  problem,  what  is  the  probability  of  drawing  black 
and  red? 

The  probabilities  of  the  (independent)  .events  are  respectively  ^ 
and  y%.     The  probability  is  then  ^. 

546.  If /i,  ^2»  /a*  •  •  •  A  Stand  for  the  respective  prob- 
abilities of  n  independent  events,  it  is  evident  that  we 
may  extend  the  results  of  the  previous  article  and  say 
Pip2pz  '  '  -A  is  the  probability  that  a// the  events  happen, 
(1— /i)(l— /2)(l-/3)  .  •  .  (1— A)  is  the  probability  that 
all  the  events  fail  to  happen,  /^  (1  —p<i)  (1  —pz)  ...  (1  — AO 
is  the  probability  that  the  first  event  happens  and  all 
the  rest  fail  to  happen,  etc.,  etc. 

The  probability  that  at  least  one  of  the  events  happen 
is  1--(1— /i)(l— /2)(1-"A)  •  •  •  (1— A)»  since  it  is  com- 
plementary to  the  case  in  which  all  the  events  fail  to 
happen. 


THEORY    OF    PROBABILITIES.  387 

(1)  If  four  coins  be  tossed,  what  is  the  probability  that  all  will 
turn  heads. 

This  problem  has  already  been  solved.     See  Art.  541,  Ex.  (3). 

Considering  the  event  in  question  as  the  concurrence  of  four  inde- 
pendent events,  we  may  say  that  the  probability  of  any  one  coin 
falling  heads  being  ^  the  probability  that  the  four  coins  fall  heads  is 
iXiXiXior^J^. 

(2)  If  four  coins  are  tossed  what  is  the  probability  that  exactly  one 
turns  heads  ? 

The  probability  that  the  first  turns  heads  is  \,  and  the  probability 
that  the  others  turn  tails  is  iXjXi  or  J.  Then  the  probability  of  the 
first  turning  heads  and  the  others  turning  tails  is  ^XJ  or  ^.  Simi- 
larly, the  probability  that  the  second  coin  turns  heads  and  the  others 
turn  tails  is  ^,  and  likewise  for  the  third  and  for  the  fourth  coins. 
Whence,  the  total  probability  of  one  and  only  one  coin  turning  heads 

(3)  If  three  dice  be  thrown,  find  the  probability  that  at  least  one 
will  turn  6. 

The  probability  that  the  first  does  not  turn  6  is  J.  Similarly,  for 
each  of  the  others.  Whence,  the  probability  that  none  turn  6  is 
|X|X|  or  \\%\  The  probability  that  this  does  not  happen,  or  that 
one  at  least  does  turn  6,  is  1  —  iff  or  ^^. 

547.  The  formulas  of  Art.  545  are  evidently  true  for 

two  dependent  events,  provided  that  — —7- stands  for  the 

probability  of  the  second  event  after  the  first  is  known  to 
have  happened, 

(1)  What  is  the  probability  of  holding  four  aces  in  a  game  of  whist? 
The  probability  of  holding  the  first  ace  is  J|.     If  that  is  held,  the 

probability  of  holding  the  next  ace  is  Jf .  If  the  first  two  are  held, 
the  probability  of  holding  the  third  is  W.  If  the  first  three  are  held, 
the  probability  of  holding  the  fourth  is  Jg.  Whence,  the  probability 
isiiXifXiJXigor^fiJ^. 

(2)  A,  B,  and  C  toss  in  succession  for  a  prize,  which  is  to  go  to  the 
one  who  first  gets  heads.     Find  their  respective  chances. 

The  probability  that  A  gets  the  prize  the  first  throw  is  \.  B's 
probability  at  the«first  throw  is  compounded  of  the  probabilities  that  A 
iails  and  that  he  himself  wins,  or  is  ^X^  or  ^.     C's  probability  at 


388  UNIVERSITY   ALGEBRA. 

his  first  throw  is  compounded  of  the  probabilities  that  A  and  B  lose 
and  he  himself  wins,  ^X^X^  or  |.  A's  probability  at  his  second 
throw  is  similarly  j^.  B's  at  his  second  throw  is  ^,  etc.  Whence^ 
we  have  the  following  total  probabilities  : 

A's=i  +  3iff+.  .  .  ;  B's=i  +  3>5  +  .  .  .  ;  Cs=i  +  ^  +  .  .  .  ; 
or,  A's=f;  B's=f;  C's=f. 

MATHEMATICAI^   EXPECTATION. 

548.  If  p  is  the  probability  of  the  occurrence  of  a  cer- 
tain event  and  m  the  sum  of  money  which  a  person  will 
realize  in  case  the  event  occurs,  then  the  sum  of  money 
expressed  by  pm  is  called  his  Mathematical  Expecta- 
tion. Thus,  suppose  a  man  is  to  receive  $5  if  a  die  falls 
ace.  Then  his  mathematical  expectation  is  $5  multiplied 
by  the  probability  (|)  of  the  event,  or  $.83^. 

The  mathematical  expectation  represents  the  average 
sum  which  would  be  realized  from  each  event  provided  a 
very  large  number  of  trials  be  made  in  which  the  given 
event  is  in  question.  For,  in  n  trials  (^n  being  a  large 
number)  the  event  will  occur,  on  the  average,  pn  times. 
The  amount  realized  each  time  the  event  happens  being 
%in,  the  whole  amount  realized  will  be  %pnm^  an  average 
of  %pm  realized  from  each  trial. 

There  is  an  important  distinction  between  tke  equitable  value  of  a 
trial  and  the  mathematical  expectation.  If  the  probability  of  the 
success  of  a  venture  be  ^gi  and  the  sum  to  be  realized,  if  successful, 
be  $1,000,000,  the  mathematical  expectation  is  $100,000,  but  one  could 
not  prudently  give  this  amount  for  the  given  chance.  If  a  person's 
capital  be  small,  it  might  be  imprudent  to  give  $500  for  the  chance. 
Even  though  the  person  were  sure  he  would  be  able  to  take  advantage 
of  identical  opportunities  on  many  occasions,  he  must  bear  in  mind 
that  there  is  sojue  probability  that  he  will  lose  every  time.  The  com- 
putation of  this  margin,  in  connection  with  the  known  size  of  his  capital 
to  start  with,  will  determine  what  a  man  might  equitably  give  for  a 
given  chance,  which  in  every  case  would  differ  from  the  mathematical 
expectation,  unless  his  capital  or  credit  be  unlimited  in  amount. 


THEORY   OF   PROBABILITIES.  389 

KXAMPI^ES. 

1.  What  is  the  probability  of  throwing  one  ace  and 
only  one  if  four  dice  be  thrown  ? 

2.  If  four  dice  are  thrown,  what  is  the  probability 
that  two  faces  at  least  turn  alike  ? 

3.  If  four  dice  are  thrown,  what  is  the  probability 
that  exactly  two  faces  turn  alike  ? 

4.  What  is  the  expectation  when  drawing  a  bill  from 
one  of  two  boxes,  one  box  containing  five  $1  and  three 
$5  and  the  other  containing  two  $1  and  four  $5? 

5.  A  person  undertakes  to  throw  5  or  6  points  in  a 
single  throw  with  two  dice.  Find  the  probability  of 
success. 

6.  If,  on  an  average,  9  ships  out  of  10  return  safe  to 
port,  what  is  the  chance  that  out  of  5  ships  expected,  at 
least  3  will  arrive  ? 

7.  A  teetotum  of  8  faces,  numbered  from  1  to  8,  and 
a  common  die  are  thrown.  What  is  the  probability  that 
the  same  number  is  turned  up  on  each  ? 

8.  A  person  goes  on  throwing  a  single  die  until  it 
turns  up  ace.  What  is  the  probability  that  he  will  have 
to  make  at  least  ten  throws  ?  that  he  will  have  to  make 
exactly  ten  throws? 

9.  Out  of  a  bag  containing  12  balls,  5  are  drawn  and 
replaced,  and  afterwards  6  are  drawn.  Find  the  proba- 
bility that  exactly  3  balls  were  common  to  the  two 
drawings. 

10.  A  purse  contains  four  silver  dollars  and  one  eagle. 
A  second  purse  contains  ten  silver  dollars.  If  two  coins 
be  taken  at  random  from  the  first  purse  and  placed  in  the 
second,  what  is  t^e  probable  value  of  the  contents  of  each 
purse? 


390  UNIVERSITY   ALGEBRA. 

11.  A  has  three  tickets  in  a  lottery  where  there  are 
3  prizes  and  6  blanks.  He  also  has  a  ticket  in  another 
lottery  where  there  is  but  1  prize  and  2  blanks.  Find  the 
probability  of  his  drawing  at  least  one  prize. 

12.  In  a  bag  there  are  5  black  and  4  white  balls.  If 
they  be  drawn  out  in  succession,  what  is  the  probability 
that  the  first  will  be  white,  the  second  black,  and  so  on^ 
alternately,  white  and  black  ? 

13.  A  bag  contains  6  black  balls  and  1  red.  A  person 
is  to  draw  them  out  in  succession,  and  is  to  receive  a 
dollar  for  every  ball  he  draws  until  he  draws  the  red  one. 
What  is  his  expectation  ? 

14.  If  four  cards  be  drawn  from  a  pack,  what  is  the 
probability  that  they  will  be  marked  one,  two,  three,  four? 
If  they  be  drawn  in  succession,  what  is  the  probability  of 
the  order  one,  two,  three,  four  ? 

15.  A  letter  is  chosen  at  random  out  of  each  of  the 
words  musical  and  a  musing.  What  is  the  proba- 
bility that  the  same  letter  is  chosen  in  each  case  ?  If  the 
letters  are  both  consonants,  what  is  the  probability  that 
they  are  the  same  ? 

16.  A  has  three  dollar  pieces  and  B  has  two.  They 
agree  that  each  shall  toss  his  money  and  the  one  who 
gets  the  greater  number  of  heads  shall  have  all  the  money. 
Is  this  a  fair  arrangement?  What  are  the  respective 
expectations  (1)  if  they  make  but  one  throw?  (2)  if  they 
throw  until  one  of  the  two  wins  ? 

17.  A  person  undertakes  in  two  throws  with  two  dice 
to  make  7  points  the  first  throw  and  9  points  the  second 
throw.  Find  the  probability  (1)  that  at  least  one  throw 
is  successful,  (2)  that  both  throws  are  successful,  (3)  that 
only  the  first  throw  is  successful,  (4)  that  both  throws 
fail,  (5)  that  one  throw  (and  no  more)  succeeds. 


THEORY   OF   PROBABILITIES.  391 

18.  A  purse  contains  four  silver  dollars  and  an  eagle, 
and  a  second  purse  contains  five  silver  dollars.  If  two 
coins  be  taken  from  the  first  purse  and  put  in  the  second, 
and  then  if  three  coins  be  taken  from  the  second  purse 
and  placed  in  the  first,  what  is  the  probable  value  of  each 
purse? 

SUCCKSSIVB   TRIAI^. 

549.  Two  Trials.  Let  there  be  an  event  which  must 
turn  out  in  one  of  two  ways,  W  or  B,  as  in  drawing  a 
ball  from  an  urn  containing  white  and  black  balls  only. 
Let  the  probability  of  W^  happening  be  p  and  of  B  hap- 
pening be  q,  so  that  p+q  must  equal  unity.  If  two  trials 
be  made  (as  drawing  from  an  urn,  restoring  the  ball,  and 
drawing  again)  the  four  possible  cases  which  may  occur 
are  WW,  WB,  BW,  BB,  the  respective 
probabilities  being /^2,      pq,     qp,     q'^, 

by  the  theory  of  compound  events.  In  other  words,  the 
probability  of  drawing  two  white  balls  is  p'^\  the  proba- 
bility of  drawing  white  and  black  in  the  order  WB  is  pq\ 
the  probability  of  drawing  black  and  white  in  the  order 
BW  is  likewise  pq;  the  probability  of  drawing  two  black 
balls  is  q^.  The  probability  of  drawing  white  and  black, 
irrespective  of  the  order  in  which  they  are  drawn,  is 
plailny  2pq,     Thus,  we  note 

550.  n  Trials.  Instead  of  two,  letthere  be  T^trials  made. 
Then  the  probability  of  W  happening  every  time  is  p"". 

The  probability  of  I^happening  n--l  times  and  B  once 
in  an  assigned  order  is  p**~^  q.  If  the  order  is  indifferent, 
the  probability  of  W  happening  n—1  times  and  B  once 
is  the  sum  of  the  probabilities  of  B  happening  on  the  first 
trial  and  PF"  happening  on  the  other  trials,  of  B  happening 
on  the  second  trial  and  Won  the  other  trials,  etc.,  or 
P'*''^q+P*'~^q+P"~'^q+'  .  .ton  terms  ^^np^'-^q. 


392  UNIVERSITY   ALGEBRA. 

The  probability  of  W  happening  n—2  times  and  B 
twice  in  an  assigned  order  is  />'*~^$'^.  If  a  specified  order 
is  not  required,  the  2  trials  on  which  B  happens  can  be 

selected  from  the  n  trials,  in  ~ — ^-  ways.    Whence,  the 

probability  of  W  happening  n—^  times  and  B  twice,  if 

n(n — 1") 
the  order  is  indifferent,  is  the  sum  of  — :j — x-  terms  each 

fi(fi '\\ 

equal  to  p**~^q^i  and  is  therefore  equal  to  —i — ~  />**""  ^^^. 

In  general,  the  probability  of  PP' happening  ^—r  times 
and  B  happening  r  times  in  an  assigned  order  is  P'~''q''. 
If  a  specified  order  is  not  required,  the  r  trials  on  which 

B  happens  can  be  selected  from  the  n  trials  in 


\7i—r 


ivays.     Whence,  the  probability  of  W  happening  n^r 
times  and  B  r  times,  if  the  order  is  indifferent,  is  the  sum 

\n 
of  -, — ^= —  terms  each  equal  to  P"  ''^'',  and  is  therefore 


L- 


n — r 


\n 

equal  to  -. — r= —  P'^<f'    Thus,  we  have  shown  that 

^  \r  \n—r  ^     ^ 

In  the  binomial  formula 


the  successive  terms  are  the  respective  prohal)ilities  of  the  event 
W happening  exactly  7i  times;  of  W happening  n—1  times  and 
B  once;  of  W  happening  n—^  times  and  B  twice  ^  and  so  on. 
It  is  well  to  note  in  the  above  that  it  does  not  matter 
whether  we  speak  oithe  happening  of  B  or  the  failing  of  IV. 
It  is  easily  seen  then  that  the  probability  of   W  failing 

\n 

exactly  r  times  in  n  trials  is  -j — ^=— ,-  p*'g**'^. 

''  m—r   \r^  ^ 


THEORY   OF   PROBABILITIES.  393 

551.  At  least  r  Successes.  The  probability  of  W 
happening  at  least  r  times  out  oin  trials  can  be  decomposed 
into  the  probabilities  of  it  happening  every  time,  every 
time  but  one,  every  time  but  two,  .  .  .  ,  exactly  r  times. 
Thus,  the  probability  of  at  least  r  successes  in  n  trials  is 

552.  The  following  examples  show  some  application 
of  the  above  and  analagous  principles  : 

(1)  What  is  the  probability  of  throwing  exactly  three  6's  in  five 
trials  with  a  single  die  ? 

Here  /=J,  ^=|,   «=5,   r=2.     Whence,  the  required  probability 

is  ^-^3  (4)^(|)^  =  3¥#8  =  .03215. 

(2)  The  odds  are  2  to  1  in  favor  of  A  winning  a  single  game  against  B. 
Find  the  probability  of  A's  winning  at  least  2  games  out  of  3. 

Here  p—^,  9~h>  ^—^>  r—^.  The  probability  required  may  be 
decomposed  into  the  probabilities  of  winning  all  3,  and  of  winning 
exactly  2  out  of  3  games.     Whence,  we  have  (l)^  4-3(|)2  |=p, 

(3)  The  odds  are  2  to  1  in  favor  of  A  winning  a  single  game 
against  B.  A  lacks  8  games  and  B  lacks  5  games  of  winning  a  match. 
Find  A's  probability  of  winning  the  match. 

A  must  win  his  8  games  out  of  the  next  12  games,  for  otherwise  B 
will  win.  Whence,  A's  probability  of  success  is  composed  of  the 
probabilities  of  his  winning  exactly  8  games  out  of  the  next  8,  of  his 
winning  exactly  8  games  out  of  the  next  9,  and  of  his  winning  exactly 
8  games  out  of  the  next  10,  ...  ,  and  of  his  winning  exactly  8  games 
out  of  the  next  12.     Altogether  this  gives 

10  .  9  11  .  10  .  9 

<i)»+9(i)« Kt:-2  (i)^(i)'+  1.2.3  ^^)°^i)' 

12.11.10.9^     ^ 
^    1.2.3.4    ^^^  ^^^ 
=(f)«(l  +  3i-5+s9«  +  5Ji):^.828. 


394  UNIVERSITY    ALGEBRA. 

MISCKI.I.ANKOUS  KXKRCISES. 

I.  What  is  the  probability  of  throwing  each  of  the 
sums  2,  3,  4,  5,  ...  ,  12  in  a  single  throw  with  two  dice? 

The  number  of  cases  in  question  is  6^  or  36.  The  number  of 
ways  of  throwing  any  required  sum,  say  8,  is  the  coeflScient  of  x^  in 
the  expansion  of  {x'^+x^+x^ +x^-{-x^+x^)^,  for  (see  generalized 
distributive  law  of  multiplication)  this  coefficient  is  the  number  of 
ways  in  which  8  can  be  made  up  by  the  addition  of  two  of  the 
numbers  1,  2,  3,  4,  5,  6.  But  (^i  +  ^^^x^-f ^<+a:5  +  «)*  = 
^2  4.  2x^  +  3^*  -4-  4x^  -4-  5^«  +  6x'r  +  5x^  +  4x»  +  Sx^^  ^  2x^^  +  x^^. 
Whence,  we  have  the  probabilities  of  the  different  throws  with  two 
dice  as  follows ; 

Throws.  2      3      4      5      6      7      8      9     10    11     12 

Probabilities,       g^     3%     ^    ^     is    /tf    /«     5%    A    5«     sV 

2.  If  four  dice  be  thrown,  what  is  the  probability  that 
they  all  fall  differently  ? 

3.  What  is  the  probability  of  throwing  15  in  a  single 
throw  with  three  dice  ? 

4.  What  are  the  odds  against  throwing  7  twice  at 
least  in  three  throws  with  two  dice  ? 

5.  In  three  throws  with  a  pair  of  dice,  find  the  proba- 
bility of  having  doublets  two  or  more  times? 

6.  What  is  the  probability  of  throwing  double  sixes 
once  or  oftener  in  three  throws  with  a  pair  of  dice  ? 

7.  Compare  the  chances  of  throwing  eight  with  two 
dice  and  twelve  with  three  dice,  having  two  trials  in  each 
case. 

8.  Find  the  probability  of  drawing  two  black  balls 
and  one  red  from  an  urn  containing  five  black,  three  red, 
and  two  white. 

9.  Two  persons,  A  and  B,  engage  in  a  game  in  which 
A's  skill  is  to  B's  as  2  is  to  3.  Find  the  probability  of 
A's  winning  at  least  2  games  out  of  3. 


THEORY    OF   PROBABILITIES.  395 

10.  Show  that  there  is  more  probability  of  throwing 
nine  in  a  single  throw  with  three  dice  than  of  throwing 
nine  in  a  single  throw  with  two  dice,  the  odds  being  25 
to  24. 

11.  A,  B  and  C  are  tied  for  a  prize  of  $50.  A  and  B 
toss  a  coin  and  the  winner  tosses  with  C,  the  final  winner 
to  have  the  prize.     Find  the  expectation  of  each  person. 

12.  A  and  B  play  at  chess,  and  A  wins  on  an  average 
two  games  out  of  three.  Find  the  probability  of  A  win- 
ning exactly  four  games  out  of  the  first  six,  drawn  games 
being  disregarded. 

13.  A  and  B  play  at  chess,  and  A  wins  on  an  average 
five  games  out  of  nine.  Find  A's  chance  of  winning  a 
majority  (1)  out  of  three  games,  (2)  out  of  four  games, 
(3)  out  of  nine  games,  drawn  games  being  disregarded. 

14.  Two  players  of  equal  skill,  A  and  B,  are  playing  a 
set  of  games.  They  leave  off  playing  when  A  wants  3 
games  and  B  wants  2  games.  If  the  stake  is  $16  what 
share  ought  each  to  take  ? 

15.  Three  white  balls  and  five  black  are  placed  in  a  bag, 
and  three  persons  each  draw  a  ball  in  succession  (the 
balls  not  being  replaced)  until  a  white  ball  is  drawn. 
Show  that  their  respective  probabilities  are  as  27,  18,  11. 

16.  Two  players  of  equal  skill,  A  and  B,  are  playing  a 
set  of  games.  A  waiits  two  games  to  complete  the  set  and 
B  wants  three  games.  Compare  the  probabilities  of  A 
and  B  for  winning  the  set. 

17.  Supposing  that  it  is  8  to  7  against  a  person  who  is 
now  30  years  of  age  living  till  he  is  60,  and  2  to  1  against 
a  person  who  is  now  40  living  till  he  is  70,  find  the  prob- 
ability that  one  at  least  of  these  persons  will  be  alive  30 
years  hence. 


CHAPTER  XXIV. 

CONVBRGKNCK  AND  DIVERGENCE)  OF  SERIES. 

553.  A  Series  is  a  succession  of  terms  which  are 
arranged  one  after  another  in  accordance  with  some  fixed 
law.  Arithmetical,  geometrical,  and  harmonical  pro- 
gressions are  examples  of  series. 

554.  A  series  in  which  the  number  of  terms  is  unlim- 
ited is  called  an  Infiinite  series,  and  one  in  which  the 
number  of  terms  is  limited  is  called  a  Finite  series. 

555.  An  infinite  series  is  Convergent  if  the  sum  of 
its  first  n  terms  approaches  a  definite  limit  as  n  increases 
indefinitely.     This  limit  is  called  the  sum  of  the  series. 

556.  An  infinite  series  is  Divergent  if  the  sum  of  its 
first  n  terms  increases  indefinitely  in  absolute  value  as  n 
increases  indefinitely. 

557.  If  the  sum  of  the  first  n  terms  of  an  infinite  series 
does  not  increase  indefinitely  nor  yet  approach  a  definite 
limit  as  n  increases  indefinitely,  the  series  is  said  to  be 
Indeterminate  or  Neutral  or  Oscillating. 

The  series  l—x-\-x^—x^-i-  ...  is  convergent  if  ^<1, 
divergent  if  :r>l,  and  indeterminate  if  x=l. 

558.  Series  are  often  used  to  replace  certain  expres- 
sions the  values  of  which  are  given  to  closer  and  closer 
degrees  of  approximation  by  taking  more  and  more  terms 
of  the  series. 

By  means  of  series  we  are  often  enabled  to  calculate 
the  numerical  values  of  expressions  more  easily  and  more 


CONVERGENCE  AND  DIVERGENCE.  39/ 

rapidly,  or  to  study  the  properties  of  functions  more  easily 
than  could  be  done  without  the  use  of  series.  But  in 
order  that  this  simplification  be  permissible  it  is  necessary 
that  the  series  substituted  should  have  the  proposed  func- 
tion for  a  limit,  i.  <?.,  that  the  series  should  be  convergent. 
The  necessity  of  using  only  convergent  series  will  be  seen 
by  an  example. 

Let  us  divide  1  by  \^x.     The  work  is  as  follows  : 
1— .^ll       \l  +  x+x^+x^+x^. 
1-x 

X 

x—x^ 


X'' 


x^ 
x^ 


X 


4_^5 


As  this  division  may  be  continued  to  any  extent  desired, 
it  appears  that 

-, =  1+^+^2+^^+^*+  •  •  • 

\  —  x 

If  we  put  -^=To  ^^  ^^^^  equation,  the  first  member  be- 
comes -y-,  and  the  second  member  becomes  1.111+  •  •  • 
which  evidently  approaches  nearer  and  nearer  the  value 
y-  as  the  expression  is  carried  out  to  a  greater  and  greate^ 
number  of  decimal  places,  /.  ^.,  as  more  and  more  terms 
of  the  series  are  taken. 

If  now  we  put  x=10  in  the  same  equation,  the  first 
member  becomes  —  ^,  and  the  second  member  becomes 
1  +  10 +  100 +  1000+  •  •  •  which  plainly  does  not  approach 

— i.     Thus   it  is  plain  that  the  fraction  -z cannot  be 

1—x 

placed  equal  to  the  series  l  +  x-^x'^+x^-^  -  •  •  when  x=  10 
even  though  the  series  was  obtained  by  actual  division. 


398  UNIVERSITY   ALGEBRA. 

If  in  the  same  equation  as  before  we  put  x=—l,  the 
first  member  becomes  ^  and  the  second  member  becomes 
1  —  1  +  1—1  +  1—1+  •  .  .  which  plainly  does  not  ap- 
proach ^.     Thus  it  is  plain  that  the  fraction  :j cannot 

be  placed  equal  to  the  series  l+x+x^+x^+x^+  .  •  • 
when  x=--l  even  though  the  series  was  obtained  by- 
actual  division.  Thus  it  appears  that,  in  general,  no  use 
can  be  made  of  divergent  or  indeterminate  series.  For 
this  reason  it  is  to  be  understood  that  whenever  infinite 
series  are  used  the  results  hold  onfy  so  long  as  the  series 
used  or  obtained  are  convergent. 

In  many  cases  a  series  is  convergent,  divergent,  or 
indeterminate  according  to  the  value  of  some  letter  in  the 
series,  and  it  is  always  understood  in  such  cases  that  the 
letter  concerned  is  limited  to  those  values  which  make  the 
series  convergent,  and  no  inference  is  to  be  drawn  for  any 
other  values, 

559.  It  would  be  fortunate  if  some  simple  and  universal 
criterion  were  known  by  means  of  which  we  could  deter- 
mine whether  any  given  series  is  convergent  or  not,  but 
unfortunately  no  such  criterion  has  been  found.  There 
are,  however,  some  cases  in  which  we  can  determine  the 
character  of  a  given  series,  and  some  of  these  will  be  given 
in  what  follows. 

560.  Let  the  terms  of  a  series  be  represented  by  Wj, 
«2j  ^3>  etc.,  in  each  case  the  subscript  being  the  same  as 
the  number  of  the  term;  and  let  -^j  be  the  remainder 
after  the  first  term,  R^  the  remainder  after  the  second 
term,  R^  the  remainder  after  the  third  term,  etc. ;  in  each 
case  the  remainder  after  any  number  of  terms  are  taken 
is  represented  by  R  with  a  subscript  equal  to  the  number 
of  terms  already  taken ;  and  further,  let  the  sum  of  any 


CONVERGENCE  AND  DIVERGENCE.  399 

number  of  terms  be  represented  by  S  with  a  subscript 
equal  to  the  number  of  terms  taken,  t.  e,y  the  sum  of  two 
terms  will  be  represented  by  6*2,  the  sum  of  three  terms 
by  6*3,  and  so  on. 

561.  With  the  notation  just  explained,  the  sum  of  a 
series  which  has  a  limited  number  of  terms  will  be  repre- 
sented by  Sg+Rg,  whether  ^  is  1  or  2  or  3  or  any  other 
number  not  exceeding  the  whole  number  of  terms  of  the 
series. 

In  an  infinite  convergent  series  Sn  approaches  a  limit 
as  n  increases  without  limit,  and  the  value  of  this  limit  is 

Sg-\-Rq, 
where  q  is  any  positive  whole  number  whatever.     It  is 
easy  to  see  in  this  case  that  R^^^  2,sn increases  without 
limit. 

In  an  infinite  divergent  series  Sn  does  not  approach 
any  limit  neither  does  R^  approach  any  limit,  and^S^H-^^ 
has  no  definite  value  at  all. 

562.  It  is  evident  that  a  series  cannot  be  convergent 
unless,  after  a  certain  number  of  terms  are  taken,  the 
successive  terms  decrease  in  absolute  magnitude,  or,  in 
other  words,  unless  ^^«^  0  as  n  increases  without  limit. 
But  while  this  is  necessary  it  is  not  sufiicient,  for  a  series 
may  be  divergent  and  still  2^^^  0  as  n  increases  without 
limit. 

Take  for  example  the  harmonic  series 

in  which  the  n  th  term  is  — »  which  evidently  approaches 

zero  as  n  increases  without  limit. 

If  the  terms  of  this  series  be  grouped  thus :  1 


4.00  UNIVERSITY   ALGEBRA 

then  in  no  group  is  the  sum  less  than  ^  and  as  there  is 
an  unlimited  number  of  groups,  the  series  evidently  does 
not  approach  any  limit,  but  increases  without  limit  as  the 
number  of  terms  increases  without  limit.  Therefore  the 
series  is  divergent. 

563.  Before  proving  the  theorems,  by  means  of  which 
the  convergence  or  divergence  of  series  is  usually  deter- 
mined, we  desire  to  show  how  the  convergence  of  some 
series  may  be  established  by  comparing  with  some  standard 
series. 

I^et  us  take  the  series 

where  the  only  restriction  placed  upon  the  successive 
terms  is  that  u^'::  0  as  «  increases  indefinitely.  With 
this  restriction  it  is  plain  that  as  n  increases  indefinitely 
the  sum  of  the  first  n  terms  appoaches  nearer  and  nearer 
the  value  u^.or,  as  we  usually  express  it,  the  sum  of  the 
series  equals  u^.  Therefore,  with  this  restriction,  the 
series  is  convergent,  and  we  are  permitted  to  write 

In  this  equation  we  may  make  any  substitution  we 
please  consistent  with  the  above  restriction,  that  Un'^  0> 
and  the  series  obtained  will  be  convergent. 

Let  us  take  u^  =  l,  u^=x,  u^=x^  etc. 

Wherein  x<l.     The  above  equation  thus  becomes 

or,  dividing  both  members  by  1—x, 

Yzr^=l  +  x-JrX^-hx^  \-x^A-  .  -  . 
Hence  the  series  l-^x-\-x^-\-x^-\-  •  .  . 

is  convergent  when  x<l. 

As  a  second  example  let  us  take 

«i=:l,  u^—\,  «3  =  ietc. 
The  above  equation  thus  becomes 


CONVERGENCE  AND  DIVERGENCE^  4OI 

or,  uniting  terms  within  parentheses, 

1_       _1_  1 

From  this  we  conclude  that  the  series 

1  1  1  1 

is  convergent. 

As  a  third  example  let  us  take 

zi^  =  l,  u^^l,  ^z-h  ^4=h  etc. 
The  above  equation  thus  becomes 

i-(i-J)+(^-4)  +  (J-*)+-  -  - 

or,  uniting  terms  within  the  parentheses, 

2  2  2. 

or,  dividing  both  members  by  2, 

1 1_       J_  1 

2-1.3'^3.5"*"5.7"^*** 
From  this  we  conclude  that  the  series 
1  1  1 


1.3^3.5"^5.7'^ 


is  convergent. 


564.  In  the  above  three  examples  we  have  obtained 
special  series  by  making  substitutions  in  the  standard 
equation 

but  we  may  sometimes  proceed  in  the  reverse  order  and 
reduce  a  given  series  to  this  standard  form  and  then  we 
shall  be  able  to  decide  about  the  convergence  or  divergence 
of  the  given  series. 

Let  us  take  the  series 

1__  1  1 

x{x-\-l)'^ {x-hl)[x-\-2)~^ {x-\-2){x+S)'^  '  '  ' 
Whatever  be  the  value  of  x  we  evidently  have 


x{x-{-l)       X       x-^1 

1       ^  1 1_ 

(x-\-l){x-^2)      x-\-l      x-\-2 

1  ^     1 1_ 

(^+2)(:»:  +  3)~;t:H-2      ;»r+3 


26  — U.  A. 


402  UNIVERSITY   ALGEBRA. 

Adding  these  equations,  member  by  member,  we  find 

1  ,+,,,1^,-.,     1      • 


But   the   series  in  the  second  member  being  of  the  form  of  our 
assumed  standard,  we  judge  that  the  sum  equals  —    Therefore, 
1  1         .  1  -1 


X       x(x+l)     {x-\-l){x+2)     {x+2){x+By 
From  this  we  conclude  that  the  series 

_J_  + I + 1 

is  convergent  for  all  values  of  x. 

565.  Many  more  examples  might  be  given,  but  these 
are  enough  to  show  that  in  many  cases  the  convergence 
of  series  may  be  established  by  simple  methods  without 
the  application  of  any  theorem  whatever.     We  now  pro-, 
ceed  to  a  more  systematic  treatment  of  the  subject.   , 

566.  Theorem  I.  If  from  any  convergent  series  the 
first  p  terms  be  erased,  the  remaining  series  will  be  con- 
vergent; and  conversely,  if  the  series  obtaiiied  by  erasing 
from  any  proposed  series  the  first  p  terms  is  convergent^  the 
series  itself  is  convergent,  provided  each  of  these  p  terms  is 
finite. 

According  to  the  definition  in  Art.  555,  a  series  is 
convergent  if  the  sum  of  its  first  n  terms  approaches  a 
definite  limit  as  n  increases  indefinitely. 

Now,  p  being  a  fixed  number,  the  first  p  terms  of  a 
convergent  series  has  a  definite  sum  which,  by  the  nota- 
tion of  Art.  560,  we  represent  by  Sp.  The  sum  of  the 
series  being  S,  the  series  which  remains  after  the  first  p 
terms  are  erased  has  a  definite  sum  S—S^,  Therefore, 
the  remaining  series  is  convergent 


CONVERGENCE  AND  DIVERGENCE.       403 

Again,  when  the  first  p  terms  of  a  given  series  are 
erased,  if  the  remaining  series  is  convergent,  the  sum  of 
the  terms  of  this  remaining  series  approaches  a  definite 
limit  as  more  and  more  terms  are  taken.  I^et  us  repre- 
sent this  limit  5 ',  then  the  sum  of  the  terms  erased  being 
represented  by  Sp,  which  is  definite  when  each  of  these 
erased  terms  is  finite,  the  sum  of  the  terms  of  the  given 
series  approaches  Sp  +  S'  as  a  limit,  and  as  this  expression 
is  definite,  the  given  series  is  convergent  by  definition 
Art.  555. 

567.  The  idea  expressed  in  theorem  I  is  often  stated 
as  follows : 

For  a  series  to  be  convergent  it  is  necessary  and  sufficient 
that  the  sum  of  the  first  q  terms  which  follow  the  first  p 
terms  approach  a  defiyiite  limit  as  q  increases  indefinitely, 

568.  Theoreni  II.  In  any  convergent  series  the  sum 
of  the  first  q  terms  which  follow  the  first  p  terms  appoaches 
zero  as  a  limit  as  p  increases  indefinitely. 

In  this  theorem  q  is  supposed  to  be  2,  fixed  number  but 
p  is  supposed  to  increase  indefinitely.  Representing  any 
convergent  series  by 

we  have  by  the  notation  of  Art.  560, 

5=  limit  S,, 
Also,  5=  limit  Sp^^. 

.*.  limit  6^+^=limit  Sp. 
.-.  limit  (5,^,-5^)=0. 
But         Sp^^—Sp=Up^^  +^/+2  +  ^iH-3+  •  •  •  +^5^+^. 
.-.  limit  (^^^4-i+^/+2+  •  •  •  +^^^)=0. 
The  expression  within  the  parenthesis  is  the  sum  of 
the  first  q  terms  which  follow  the  first/  terms  of  the  given 
series.     Hence  the  theorem  is  proved. 


404  UNIVERSITY   ALGEBRA. 

569.  It  should  be  noticed  that  this  theorem  states 
something  which  is  true  of  every  convergent  series,  but 
it  must  not  be  imagined  that  every  series  for  which  this 
statement  is  true  is  therefore  convergent. 

The  statement  that  the  sum  ot  the  first  q  terms  which 
follow  the  first  p  terms  approaches  zero  as  a  limit  as  p 
increases  indefinitely  is  true  of  all  convergent  series,  and 
it  is  also  true  of  some  series  which  are  not  convergent. 

570.  Theorem  III.  A  series  which  contains  positive 
and  negative  terms  is  convergent  if  the  series  obtained  by 
making  all  its  terms  positive  is  convergent. 

Let  the  limit  of  the  sum  of  the  positive  terms  be  repre- 
sented by  U^  and  the  limit  of  the  sum  of  the  absolute 
values  of  the  negative  terms  by  6^2  \  then  the  sum  of  the 
series  is  U-^  —  U^,  and  if  it  can  be  shown  that  U^  —  6^2  is 
a  definite  sum  it  will  follow  that  the  series  is  convergent. 

Now  consider  a  new  series  formed  from  the  given 
series  by  making  all  its  terms  positive.  The  sum  of  this 
new  series  is  plainly  U^^  +  U^  and  as  this  new  series  is 
convergent  by  hypothesis  U-^  +  ^2  ^^^  ^  definite  value. 
Again,  as  17^  and  6^2  ^^^  both  positive,  and  as  their  sum 
has  a  definite  value,  therefore  each  of  these  quantities 
has  a  definite  value,  therefore  their  difierence,  6^1  —  6^2 
has  a  definite  value.     Therefore,  the  series  is  convergent. 

571.  A  series  which  is  convergent  when  all  its  terms 
are  taken  positively  is  called  an  Absolutely  Convergent 
Series. 

Of  course,  any  convergent  series  whose  terms  are  all 
positive,  is  an  absolutely  convergent  series.  It  will  be 
seen  later  that  there  are  convergent  series  whose  conver- 
gence depends  on  the  fact  that  some   of  its   terms  are 


CONVERGENCE  AND  DIVERGENCE^  405 

positive  and  some  negative,  and  which  becomes  divergent 
when  all  its  terms  are  made  positive.  Such  series  are 
called  Semi-convergent  Series. 

572.  Theorem  IV.  If,  beginning  with  any  term  of 
an  absolutely  convergent  series,  all  the  subsequent  terms  be 
multiplied  by  any  finite  numbers  whatever,  positive  or 
negative,  the  resulting  series  will  be  absolutely  convergent. 

Let  us  represent  the  given  absolutely  convergent 
series  by 

^1+2^2+^3+^4+  •   •   •  (1) 

then,  taking  A,  B,  C,  etg.  to  represent  any  positive  or 
negative  finite  numbers,  let  us  multiply  the  (/)  +  l)st 
term  by  A,  the  (/>+2)d  term  by  B,  etc.  We  thus  form 
a  new  series 

2^1+2/2+  •  •  •  -\-Up-\- AUfj^^-^ Bup^^-\-  ...  (2) 

We  are  to  prove  that  the  series  (2)  is  an  absolutely 
convergent  series. 

From  the  series  (1)  form  another  series  whose  terms 
are  all  positive.     Represent  this  new  series  by 

z;i+z;2+^3+^4+  •  •  •  (S) 

Plainly,    the   terms   of    (1)    and    (3)    have   the    same 

absolute  values,  and  the  only  difference  between  these 

two  series  is  that  the  terms  of  (3)  are  all  positive,  while 

the  terms  of  (1)  are  positive  or  negative  at  pleasure. 

Also,  from  the  series  (2)  form  another  series  whose 
terms  are  all  positive,  and  represent  this  new  series  by 

^1+^2+  •    •   •  +^/  +  ^^/-M+^^/+2+-   •   •  (4) 

Plainly,  the  only  difference  between  (2)  and  (4)  is  that 
the  terms  of  (4)  are  all  positive,  while  the  terms  of  (2) 
are  not  necessarily  all  positive. 


406  UNIVERSITY    ALGEBRA. 

Now  suppose  K  to  stand  for  some  positive  finite  num- 
ber as  great  as  the  numerically  greatest  of  the  chosen 
multipliers  A,  B,  C,  etc.,  then 

But  by  theorem  I 

approaches  some  definite  limit. 

Therefore,  ^(z/^i  +^/+2  +  •  •  • )  approaches  some  defi- 
nite limit. 

Therefore,  Avp^-^  +^i>+2+  *  •  •  approaches  some  definite 
limit. 

Therefore,  v^+V2+  •••  +v^+Avj^i+Bv^2+  '  — 
which  is  the  series  (4),  is  convergent. 

Since  (4)  is  a  convergent  series,  all  of  whose  terms  are 
positive,  it  follows  that  (2)  (which  becomes  (4)  when  all 
its  terms  are  taken  positively)  is  an  absolutely  convergent 
series  by  definition,  Art.  571. 

573.  Theorem  V.  Any  series  is  absolutely  convergent 
ify  after  some  particular  term,  each  of  its  subsequent  terms 
is  numerically  less  than  the  corresponding  term  of  a  series 
which  is  known  to  be  absolutely  convergent. 

This  follows  immediately  from  the  preceding  demon- 
stration by  taking  the  multipliers  A,  B,  C,  etc.,  any 
fractions  numerically  less  than  unity. 

574.  Theorem  VI.  A  series  is  absolutely  convergent 
if,  after  any  particular  term,  each  of  its  subsequent  terms 
bears  a  finite  ratio  to  the  correspondifig  term  of  a  series 
which  is  known  to  be  absolutely  convergent. 

This  theorem  is  an  immediate  consequence  of  theo- 
rem IV.     For,  after  the/>  th  term  of  each  series,  the  ratios 


CONVERGENCE  AND  DIVERGENCE.  407 

of  the  terms  of  the  second  series  to  the  corresponding 
terms  of  the  first  series  are  Ay  B,  C,  etc.,  which  by- 
supposition  are  finite  numbers. 

575.  Theorems  IV,  V,  VI,  hold  for  semi-convergent 
as  well  as  absolutely  convergent  series,  provided,  in  each 
case,  the  terms  of  the  second  series  have  the  same  signs 
as  the  corresponding  terms  of  the  first  series. 

576.  A  series  all  ol  whose  terms  are  positive  must  be 
either  divergent  or  absolutely  convergent. 

Since  all  the  terms  are  positive,  the  more  terms  there 
are  taken  the  greater  is  the  sum  of  those  terms,  that  is, 
the  sum  of  the  first  n  terms  continually  increases  as  n 
increases.  Now,  in  this  continual  increase,  the  sum  of  the 
first  n  terms  must  increase  indefinitely  or  else  must 
approach  some  definite  limit  as  n  increases  indefinitely. 
In  the  first  case  the  series  is  divergent,  and  in  the  second 
case  the  series  is  convergent,  and,  having  all  its  terms 
positive,  is  absolutely  convergent. 

577.  Theorem  VII.  If,  after  any  particular  term  of 
a  divergent  series  all  of  whose  terms  are  positive,  all  the 
subsequent  terms  be  ?nultiplied  by  any  positive  finite  numbers 
whatever,  the  resulting  series  is  divergent. 

Let  us  represent  the  given  divergent  series  all  of 
whose  terms  are  positive  by 

2^1+2^2+^3+^4+  •   •  •  (1) 

Then  taking  A,  B,  C,  etc.,  to  represent  any  positive 
finite  numbers,  let  us  multiply  the  (/>+l)st  term  by  A^ 
the  (/+2)d  term  by  B,  etc.     We  thus  form  a  new  series 

«l+«2+^3+-   •  •+^/  +  ^^/+l+^2^i4.2+  •   •  •         (2) 

all  of  whose  terms  are  positive. 

We  are  to  prove  that  the  series  (2)  is  divergent. 


408  UNIVERSITY   ALGEBRA. 

Suppose  K  to  stand  for  some  positive  number  as  small 
as  the  smallest  of  the  chosen  multipliers  A,  B,  C^  etc., 
then 

But  the  given  series  (1)  being  divergent 

^(^/+l+«/+2+^/+3H ) 

increases  indefinitely  as  the  number  of  terms  increases 

indefinitely. 

Hence  ^?^i»+i  +Bup^^  +  Cup^^  +  •  •  • 

increases  indefinitely  as  the  number  of  terms  increases 

indefinitely.     Therefore,  the  series  (2)  is  divergent. 

578.  Theorem  VIII.  A  series,  all  of  whose  terms  are 
positive,  is  divergent  if,  after  any  particular  term,  each  sub- 
sequent term  is  greater  than  the  corresponding  term  of  a 
series  all  of  whose  terms  are  positive  and  which  is  known 
to  be  divergent. 

This  follows  immediately  from  the  preceding  demon- 
stration by  taking  the  multipliers  A,  B,  C,  etc.,  any 
positive  numbers  greater  than  unity. 

579.  Theorem  IX.  A  series,  all  of  whose  terms  are 
positive,  is  divergent  if,  after  any  particular  term,  each  of 
its  subseque7it  terms  bears  a  finite  ratio  to  the  corresponding 
tervi  of  a  series  all  of  whose  terms  are  positive  and  which  is 
known  to  be  divergent. 

This  is  an  immediate  consequence  of  theorem  VII. 
For  after  the  /  th  term  of  each  series  the  ratios  of  the  terms 
of  the  second  series  to  those  of  the  first  series  are  A ,  B^ 
C,  etc.,  which,  by  supposition,  are  finite  numbers. 

580.  Theorem  X.  If,  after  any  particular  term  of  a 
series  all  of  whose  terms  are  positive,  the  ratio  of  each  term 
to  the  preceding  is  less  than  some  fixed  number  which  is 
itself  less  than  unity  the  series  is  absolutely  convergent. 


CONVERGENCE  AND  DIVERGENCE.       409 

Let  the  series  all  of  whose  terms  are  positive  be  repre- 
sented by 

and  suppose  that  after  the  n  th  term  the  ratio  of  each 
term  to  the  preceding  is  less  than  k  where  k  is  some  fixed 
number  less  than  unity. 
We  then  have 

!^!f±i<;^,     "^^O^,     '^^<^,  etc. 

From  these  inequalities  we  readily  obtain 

^u+  2  "^  ^^^H+ 1         •  *  •  ^«+  2  *^  ^  ^  ^«' 


Thus  we  see  that,  after  the  n  th  term,  each  term  of  the 

series        7^l4-^^2+  •  •  •  -{-^^n  +  ^^n+i+^u+2+  ■  •  • 
is  less  than  the  corresponding  term  of  the  series 

But  this  last  series  is  easily  seen  to   be  convergent. 
For  when  k<,l  the  series 

is  convergent  and  its  sum  is  j— --   (see   first   example 
tinder  Art.  563).     Therefore  the  series 

is  convergent  and  its  sum  is  t^H  •     Therefore,  the  series 

^i+?^2+  •  •  •  4-^„+^^^«+/^2^n+  •  •  . 
is  convergent. 

Therefore,  by  theorem  V,  the  given  series 

u^-^U2-\-u^-\-  •  .  .  +^^«+^^«+l4-^^„+2+  •  •  • 
is  convergent,  and,  having  all  its  terms  positive,  is  abso- 
lutely convergent. 


4.IO  UNIVERSITY   ALGEBRA. 

581.  Theorem  XI.  If ^  after  any  particular  term  of  an 
infinite  series  all  of  whose  terms  are  positive,  the  ratio  of 
mch  term  to  the  preceding  is  greater  than  some  fixed  num- 
ber which  is  itself  greater  than  unity,  the  series  is  divergent. 

Representing  by  k  some  fixed  number  greater  than 
unity  and  otherwise  adopting  the  notation  of  the  preced- 
ing article,  we  have 


^«+l^^          ^«+2 

>k,      ''^+^»^,  etc. 

U^  ^  ^'      z^„+i 

^■^n+2 

From  these  inequalities  we  readily  obtain 

u,,^^>ku„. 

Z^,+  2>>^^«+l 

.-.  u„^^>k''-u„. 

^«+  3  ^  ^^«+  2 

.'.  u,+  s>k^u„. 

Thus  we  see  that  after  the  n  th  term  each  term  of  the 
series 

U^-^U2+  '  •  •  +«n  +  w«+i+  •  •  • 

is  greater  than  the  corresponding  term  of  the  series 
u^+U2-\-  •  •  •  +u„-{-ku,,-\-k'^u„-i-  .  .  . 
But  this  last  is  easily  seen  to  be  a  divergent  series. 
Por  when  >^>  1  the  series 

1+^+^2 +>^*+- • - 
is  evidently  divergent. 
Therefore,  the  series 

Un-\-kUn-\-k'^Un+k'^Un+  •  •  • 
is  divergent. 
Therefore,  the  series 

2^1+2^2  +  ^3+  •    •    •  -\-Un'\'kUn-\-k'^U^-\-  '   .  • 

is  divergent. 
Therefore,  by  Art.  678,  the  series 

^1+^2+^8+  •   •    *  +2^«+2^n+l+  •   •  • 

is  divergent. 


CONVERGENCE  AND  DIVERGENCE.  4II 

582.  Theorem  XII.     In  an  infinite  series  all  of  whose 
terms  are  positive  if,  as  n  increases  indefinitely^  the  ratio 

-^^-^  approaches  a  definite  limit  a,  the  series  is  absolutely 

convergent  if  a<X  cL'i'id  is  divergent  if  a^\. 

In   the  first  case  the  ratio  -^^  approaches  a  limit  a, 

which  is  less  than  unity.  It  may  be  that  for  some  values 
of  n  this  ratio  will  be  even  greater  than  unity,  but  as  the 
ratio  continually  approaches  a,  which  is  less  than  unity, 
it  is  plain  that  we  can  select  some  number,  say  k,  between 
1  and  a  such  that  by  taking  n  sufficiently  large,  the  ratio 

-^^^  will  become  and  remain  less  than  k,  which  being  a 

fixed  number  less  than  unity,  shows  by  theorem  X  that 
the  series  is  absolutely  convergent. 

In  the  second  case  the  ratio  -^^  approaches  a  limit  a, 

which  is  greater  than  unity.  It  may  be  that  for  some 
values  of  71  this  ratio  will  be  even  less  than  unity,  but  as 
this  ratio  continually  approaches  a,  which  is  greater  than 
unity,  it  is  plain  that  we  can  select  a  number,  say  k^ 
between  1  and  a,  such  that  by  taking  n  sufficiently  large 

the  ratio  -^^  will  become  and  remain  greater  than  k. 
u„ 

which  being  a  fixed  number  greater  than  unity,  shows 
by  theorem  XI,  that  the  series  is  divergent. 

583.  Theorem  XIII.  hi  aji  infinite  series  all  of  whose 

terms  are  positive  if  the  ratio    ''"^^  approaches  a  limit  unity 

as  n  increases  indefinitely,  then  the  series  is  divergent,  if  the 

ratio  -^^  remains  always  greater  than  its  limit,  and  may 

be  eithe?  convergent  or  divergent  if  the  ratio  ^^  remains 

u„ 

always  less  than  its  limit. 


412  UNIVERSITY   ALGEBRA. 

First.     If  the  ratio  -^^  remains  greater  than  1  then 

each  term  of  the  series  is  greater  than  the  preceding 
term,  therefore  each  term  is  greater  than  the  corresponding 
term  of  a  series  all  of  whose  terms  are  the  same,  and  since 
a  series  all  of  whose  terms  are  the  same  is  evidently  diver- 
gent, therefore  by  theorem  VIII  the  series  considered  is 
divergent. 

Second.     If  the  ratio  -^^    approaches  1  but  remains 

less  than  1  then  all  we  know  about  the  series  is  that  each 
term  is  less  than  the  preceding  term,  but  the  sum  of  the 
first  n  terms  may  approach  a  definite  limit  or  may  increase 
indefinitely  as  n  increases  indefinitely,  therefore  the  series 
considered  may,  in  this  case,  be  either  convergent  or 
divergent. 

584.  The  last  four  theorems,  which  are  very  useful  in 
determining  the  character  of  series,  may  be  stated  as 
follows : 

When,  in  a?i    infinite   series   all    of   whose    tej^ms   are 

positive,  the  ratio     ""'"^   approaches  a  definite  Ihnit  a,  as  n 

increases  indefijiitely  the  series  is  convergent  or  divergent 
accordhig  as  this  limit  is  less  than  1  or  greater  than  1; 
but  when  this  limit  equals   1    the  series   is   diverge7it  if 

the  ratio     "'^^  remains  continually  greater  than  its  liw.it  \^ 

and  may  be  either  convergent  or  divergent  when  this  limit 
remains  continually  less  than  1. 

585.  Theorem  XIV.  If,  aftef  any  particular  term 
of  any  infiiiite  series,  the  ratio  of  each  term  to  the  preceding 
is  less  than  the  ratio  of  correspojiding  terms  of  a  series 
which  is  known  to  be  absolutely  convergent,  the  given  series 
is  absolutely  convergent. 


CONVERGENCE  AND  DIVERGENCE.  413 

Let  the  given  series  be  represented  by 

^1+^2+^3+  •  •  •  +^«+^«+i+  •  •  • 
and  the  series  known  to  be  absolutely  convergent  by 

^1+^2  +  ^3+  •  •  •  +^H+^«+i+  •  •  • 
then  we  have  given  that 

^n+l  ^  ^«+l  ^^n-i-2  ^  ^«4-2  ^«+3  ^  ^«+3  ^^^ 

U„  Vn    '        2^«+i  ^«+i'        U^+l  ^«+2 

or,  writing  these  inequalities  in  another  way, 

^«+l  ^»  ^«+2  ^«+l  ^«+8  ^«+2 

From  these  inequalities  we  readily  obtain 


Now,  since  it  is  given  that  the  series 

is  absolulely  convergent,  therefore  by  theorem   IV  the 
series     v^+V2+  •  •  •  -\-v„-{-~v,,^^-\ — -v,,^2+  •  •  • 
is  absolutely  convergent,   therefore   by    theorem   I  the 
series  ~^«+  3  +  ~^«+  2  +  --  ^«+  3  +  •  •  • 

^«  ^n  ^n 

is  absolutely  convergent,  therefore  the  series 

Un  Ifn  ^n 

^n  ^n  ^n 

is  absolutely  convergent,   therefore  by  theorem  V  the 
series 

^1+^2+^3+  •  •  •+^«+2^«+i+2^«+2+^«+3+  •  •  • 
is  absolutely  convergent. 


414  UNIVERSITY    ALGEBRA. 

586.  Theorem  XV.  If,  after  any  particula?  term  of 
uny  infinite  series,  the  ratio  of  each  term  to  the  preceding  is 
greater  than  the  ratio  of  the  corresponding  terms  of  a  series 
which  is  known  to  be  divergent,  the  given  series  is  divergent. 

With  the  same  notation  as  in  the  preceding  article,  we 
have  here  given,  that 

l^n  ^«  ^^«4-l  ^«+l        ^«+2  ^«+2 

or,  writing  these  inequalities  in  another  way, 

^n+\^'^^n^         ^«4-2^^«+l^        ^W3^^«+2^    ^^^ 

^„^.l  V^        V^^^         Vn^^        V„^^         V^^^' 

From  these  inequalities  we  readily  obtain 

^«+2  >  r^^«+2       .  •.       ^«+2  >  ;rV2  . 
Vn-\-  2  ''w 

Now,  since  it  is  given  that  the  series 

is  divergent,  therefore  by  theorem  VII  the  series 
u  u  u 

is  divergent,  therefore  the  series 

is  divergent,  therefore  the  series 

u  u  %c 

Vn  ^n  *^n 

is  divergent,  therefore  by  theorem  VIII  the  series 
Is  divergent. 

587.  In  all  the  discussion  of  series  given  thus  far  it 
has  been  assumed  that  the  terms  of  a  series  are  arranged 
in  a  given  order  and  this  order  has  not  in  any  case  yet 


CONVERGENCE  AND  DIVERGENCE.  415 

given  been  changed  at  all.  As  a  matter  of  fact,  however, 
the  order  01  the  terms  of  a  series  is  sometimes  quite 
important,  for  it  has  been  shown  by  Lejeune-Dirichlet 
that,  for  a  certain  class  of  series,  a  change  in  the  order 
of  terms  may  change  the  sum  of  a  series  or  may  even 
change  the  character  from  semi-convergent  to  divergent. 

588.  To  show  the  effect  of  changing  the  order  of 
terms  of  a  semi-convergent  series,  we  shall  take  the  series 

1 l_Ll l_Ll 1_L 

and  show  that  it  is  semi-convergent  and  therefore  has  a 
definite  sum.  Then  we  shall  show  that  when  the  terms 
are  written  in  the  order 

i+i-i+i+4-i+ • •  • 

the  sum  is  not  the  same  as  before. 

589.  Let  us  now  study  the  character  of  the  series 

i-i+i-i+i-i+  •  •  •  (1) 

We  may  write 

1_^         1_1 1_        1     1_    1 

2~1  .2'       343.4'       5     6"~5  .  6'  ^*'^- 

and  thus  the  given  series  (1)  may  be  replaced  by  the 

series  ^  + J^  + J_  +  ^g_+ .  .  .  (2) 

But  the  terms  of  the  series  (2)  are  less  than  the  corres- 
ponding terms  of  the  series 

1      ,  JL_  4.  _i_  .  _J__  , 

which  has  already  been  shown  to  be  convergent.  (See 
second  example  under  Art.  563.) 

Now  since  (3)  is  convergent,  therefore  by  theorem 
V  (2)  is  also  convergent,  and,  since  (2)  is  made  by 
simply  grouping  the  terms  of  (1)  in  sets  of  two,  therefore 
(1)  is  convergent. 


4l6  UNIVERSITY   ALGEBRA. 

If  all  the  terms  of  (1)  are  raade  positive,  we  get  the 
harmc  ^  series  which  has  been  shown  to  be  divergent. 
Therefore  (1)  is  a  convergent  series  which  becomes 
divergent  when  all  terms  are  made  positive,  /.  e,,  (1) 
is  a  semi-convergent  series. 

590.     Since  the  series 

i-*+i-i+i-i+  •  •  •  (1) 

is  semi-convergent  it  has  a  definite  sum  which  we  may 
represent  by  5*. 

Let  us  represent  the  sum  of  the  series 

l+i-i+i+|-i+  •  •  •  (2) 

by  5',  and  if  we  can  prove  that  5'  has  a  definite  value  it 
will  follow  that  the  series  (2)  is  convergent,  and  if  this 
definite  value  is  different  from  6*  it  will  follow  that  a 
change  in  the  order  of  terms  has  an  effect  on  the  sum  of 
the  series. 

Now  let  us  subtract  the  series  (1)  from  the  series  (2), 
arranging  the  work  in  such  a  way  that  the  order  of  terms 
in  each  series  shall  remain  unchanged,  but  at  the  same 
time  certain  gaps  shall  be  left  in  series  (2)  to  allow  the 
terms  having  odd  denominators  in  the  series  (1)  to  come 
immediately  under  the  terms  of  (2)  which  have  the 
same  denominators,  and  letting  terms  having  even  denom- 
inators come  where  they  will.  We  have,  then, 

...  5'-5=      \      -\      +i      -i      +... 
By  inspection  we  see  that  the  right-hand  member  of 
the  equation  below  the  line  is  exactly  one-half  of  the 
series  (1),  from  which  it  is  evident  that 

Hence,  ^'=f^. 


CONVERGENCE  AND  DIVERGENCE.  417 

Thus  we  see  that  S^  has  a  definite  value  differe"^  from 
S,  from  which  we  conclude  that  the  series  (2)  is  conver- 
gent, and  that  the  change  in  the  or-^^-^  of  the  terms  of  (1) 
has  changed  the  sum  of  the  series. 

591.  Theorem  XVI.  In  an  absolutely  convei^gent series 
the  order  of  the  terms  may  be  changed  in  any  manner  with- 
out affecting  the  sum  or  the  character  of  the  series,  provided 
each  term,  whose  positio7i  is  changed  is  removed  only  a  finite 
number  of  steps  from  the  position  it  originally  occupied. 

Let  any  absolutely  convergent  series  be  represented  by 
u^-{-u^-\-u^-\-  ...  (1) 

and  let  the  series  obtained  by  making  all  the  terms  of  (1) 
positive  be  represented  by 

v^-{-v^+v^Ar  ...  (2) 

and  finally,  let  the  series  obtained  by  changing  the  order 
of  terms  of  (1)  be  represented  by 

w^-k-w^^rw^-k-  .  .  ^  (3) 

Let  Wr  represent  the  sum  of  the  first  r  terms  of  (3) 
and  Un  the  sum  of  the  first  n  terms  of  (1).  Plainly,  then, 
we  may  take  r  so  great  that  Wr  will  include  the  sum  of 
the  first  n  terms  of  (1)  and  the  sum  of  r—n  other  terms 
of  fl)  after  the  first  n  terms.  Plainly,  also,  we  may  take 
p  to  represent  a  number  so  large  that  the  r—n  other  terms 
of  (1)  will  be  found  scattered  around  between  the  terms 
Un  and  UnJ>^p.  Hence,  the  sum  of  these  r—n  other  terms 
cannot  be  greater  than 

Hence,      ^r— ^«>*^«+^«+i+z'„+2+  ■  •  •  +z^„+^ 

*The  symbol  ^  is  used  to  express  the  fact  that  the  expression 
written  on  the  left  of  the  symbol  is  not  greater  than  the  expression 
written  on  the  right.  Similarly,  the  symbol  5C  shows  that  the  expres- 
sion on  the  left  is  not  less  than  the  expression  on  the  right. 

27~U.  A. 


41 8  UNIVERSITY   ALGEBRA. 

Now,  if  we  let  n  and  therefore  r  increase  indefinitely, 
the  expression  to  the  right  of  the  sign  >  approaches  zero 
and  therefore  Wr—  U^  also  approaches  zero. 
Hence,  limit  (  Wr—  ^«)=0. 

Hence,  limit  ^^— limit  17„=0, 

Hence,  limit  W^^=  limit  17„. 

Hence,  the  sum  of  the  series  (3)  equals  the  sum  of  the 
series  (1).  Therefore,  changing  the  order  of  terms  has 
not  changed  the  sum.  ' 

If  now  we  represent  by 

2-1+^2+^8+-  •  •  (4) 

the  series  obtained  by  making  all  the  terms  of  (3)  positive, 
then  plainly  (4)  is  obtained  from  (2)  in  exactly  the  same 
way  that  (3)  is  from  (1).  Hence,  by  the  demonstration 
just  given,  the  sum  of  the  series  (4)  is  the  same  as  the 
sum  of  the  series  (2).  Therefore,  (4)  is  a  convergent 
series  all  of  whose  terms  are  positive,  and  since  (4)  is 
the  series  obtained  by  making  all  the  terms  of  (3)  posi- 
tive and  since  (4)  is  convergent,  therefore  (3)  is  an 
absolutely  convergent  series.     Therefore,  a  change  in  the 

,  order  of  the  terms  Of  (1)  has  not  changed  either  the  sum 

•  or  the  character  of  the  series. 

592.  Theorem  XVII.  If  the  terms  of  a  series  are 
alternately  positive  and  negative  and  after  some  particular 
term,  each  term  is  numerically  less  than  the  preceding  one^ 
and  the  nth  term  approaches  zero  as  n  increases  indefinitely ^ 
the  series  is  convergent. 

Let  the  series  be 

and  let  the  sum  of  the  series  be  represented  by  S\  then, 
with  the  notation  of  Art.  560,  we  may  write  either 

6'=5^+(a^^-fi-^^+2)  +  K+8-^^+4)+  •     •         (1) 
or        ^=  Vi-K+2-^^+3)-K+4-«i^+5) C2) 


CONVERGENCE  AND  DIVERGENCE.  419 

After  a  certain  number  of  terms,  say  k,  each  term  is 
less  than  the  preceding  one,  so  if  q  be  chosen  larger  than 
k,  each  parenthesis  in  (1)  and  also  in  (2)  is  positive,  and 
therefore  from  (1)  S^Sq, 

and  from  (2)  S<^Sq^^. 

Thus  we  see  that  kS*  is  intermediate  in  value  between 
Sq  and  Sqj^^,  which  two  quantities  differ  by  ^^+1. 

Similarly,  whatever  positive  whole  number  be  repre- 
sented by  Ty  we  get        ►S>6'^+2^ 
and  ►S'<KS'^+2r+i. 

Now  Sq^^r^Sq 

and  5^+2r+i<KS^+i. 

Therefore  6*  is  intermediate  in  value  between  two 
quantities,  the  larger  of  which  grows  smaller  and  the 
smaller  of  which  grows  larger. 

Moreover,  Sq^^r^^  and  kS^+2^  differsby  2^^+ 2^+1,  which 
approaches  zero  as  r  increases;  therefore  the  two  quanti- 
ties between  which  6*  is  always  found  approach  equality 
as  r  increases.  Therefore  kS  has  a  definite  value,  or,  in 
other  words,  the  series  is  convergent. 

This  demonstration  simply  proves  this  series  to  be 
convergent.  Sometimes  the  series  will  be  absolutely 
convergent  and  sometimes  only  semi -convergent. 

593.  As  it  is  sometimes  necessary  to  consider  the  con- 
vergence of  a  series  obtained  by  multiplying  two  series 
together  the  following  theorem  is  introduced : 

Theorem  XVIII.     If  the  series 

is  absolutely  convergent  and  the  series 

^1+^2+^3  +  ^4+  •  •  •  (2) 

is  absolutely  convergent  or  semi- convergent,  then  the  series 

^l^l+(^1^2+«2^l)  +  (?^\^3+^2^2+^8^l)+  •    •   •         (3) 

which  is  formed  by  multiplying  the  series  (1)  andijL)  together 
's  convergent. 


420  UNIVERSITY   ALGEBRA. 

When  the  series  (1)  and  (2)  are  multiplied  together  we 
group  the  terms  of  the  product  as  indicated  by  the  paren- 
theses in  the  series  (3),  i.  e.,  the  sum  of  those  terms  in 
which  the  sum  of  the  subscripts  is  3  is  to  be  taken  for  the 
second  \,^rvcL  of  the  series  (3),  the  sum  of  those  terms  in 
which  the  sum  of  the  subscripts  is  4  is  to  be  taken  for  the 
/^/r<f  term  of  (3),  and  so  on. 

lyCt  Un  denote  the  sum  of  the  first  n  terms  of  (1),  V^ 
the  sum  of  the  first  n  terms  of  (2),  W^  the  sum  of  the  first 
n  terms  of  (3).  A  careful  inspection  of  the  product 
obtained  by  multiplying  the  sum  of  the  first  n  terms  of 
(1)  by  the  sum  of  the  first  n  terms  of  (2)  will  show  that 
the  product  equals  the  sum  of  the  first  n  terms  of  (3) 
plus  other  terms  in  which  the  sum  of  the  subscripts  is 
more  than  ^  +  1  but  not  more  than  In. 
'  Therefore,  [/„V,,=  W„+J^„ 
where  I^„=U2V„+u^v„_i+u^v^_2+  •  •  •  +^«^2 


Hence  7?„=e^2^«+^3(^«-i +^«)  +  ^4(^«-2 +  ^«-i+^«)  + ••- 

+  U„(^V2+'^8+  •    •   •  +^«)- 

Now  in  this  expression  for  ^«  it  is  to  be  noticed  that 
each  of  the  expressions  within  the  parentheses  is  the  sum 
of  a  certain  number  of  terms  of  the  convergent  series  (2) 
and  the  terms  occur  in  the  same  order  as  in  the  given 
series.  Therefore  none  of  the  expressions  within  these 
parentheses  can  increase  indefinitely;  /.  e.j  these  all  have 
finite  values. 

Now  if  we  let  n  increase  indefinitely,  the  expression 
for  i?«  becomes  a  series  which  is  obtained  by  multiplying 
the  terms  of  the  absolutely  convergent  series  (1)  (after 


CONVERGENCE  AND  DIVERGENCE.  42] 

the  first  term)  by  finite  numbers  and  hence  by  theorerc 
IV,  the  terms  of  Rn  constitute  an  absolutely  convergeni 
series.  Hence,  Rn  approaches  a  definite  value  as  n  in- 
creases indefinitely.  But  because  (1)  and  (2)  are  con- 
vergent series,  therefore  Un  and  FJ|  approach  definite 
limits  as  n  increases  indefinitely,  and  therefore  the  product 
UnVn  approaches  a  definite  limit  as  n  increases  indefi- 
nitely. 

Now,  since  J^«=  C/«  V^—Rn  and  since  each  term  of  the 
second  member  has  been  shown  to  approach  a  definite 
limit  as  n  increases  indefinitely,  therefore  W^  must  ap- 
proach a  definite  limit  as  n  increases  indefinitely,  therefore 
the  series  (3)  is  convergent. 

Corollary.  If  the  series  (1)  and  (2)  are  both  absolutely 
convergent,  then  the  series  (3)  is  absolutely  convergent. 

Let  the  series  formed  by  making  all  the  terms  of  (1) 
positive  be  ^1+^2 +^3 +-^4+  •  *  *  (4) 

and  the  series  formed  by  making  all  the  terms  of  (2) 
positive  be  yx+y^-^yz  +J^4  +  •  •  •  (5) 

then  (4)  and  (5)  being  absolutely  convergent,  the  above 
demonstration  shows  that  the  series 

formed  by  multiplying  (4)^and  (5)  together  is  convergent. 
Now,  since  the  terms  of  (1)  have  the  same  absolute  values 
as  the  corresponding  terms  of  (4),  and  the  terms  of  (2)  the 
same  absolute  values  as  the  corresponding  terms  of  (5), 
therefore,  whatever  the  signs  of  the  terms  of  (1)  and  (2), 
no  term  of  (3)  can  be  greater  in  absolute  value  than  the 
corresponding  term  of  (6).  Therefore,  whatever  the 
combination  of  signs  the  sum  of  the  terms  of  (3)  cannot 
exceed  the  sum  of  the  terms  of  (6);  therefore,  (3)  is  an 
absolutely  convergent  series. 


422  UNIVERSITY   ALGEBRA. 

EXAMPI^KS. 

Determine  the  character  of  the  following  infinite  series: 

^*   ^  '  "2  ~^3"2   '  43"^  *  '  * 

12      23      34 

^•^  +  [2-+|3-+^+--- 

3.    1+I  +  — +  |3+-   •   •Wll^^^<l- 

5. ; 1 --^ rQ-+  •  •  -when  x  and  a  are 

X    x+a     x-\-2a     x+6a 


positive. 


/y  ^"  yyS  '♦"4 


± 1_ 1 1_ 

7-  ^j,/     (^x+lXj'+iy  (x+2Xj'+2)     Cx+SXy+B) 
•  when  j;  and  j/  are  positive. 


(;r+4)(j'+4) 


-*/•  '^"2  <y-8  "^K*^ 


22      33      44 
9-  1+ |2  +  |J+|4-+ 


10. 


v1+Vl+ vl+VI+ 


CONVERGENCE  AND  DIVERGENCE.  423 


12.     1  +  Q2    '    QS""    44"'     * 


_j_       -j.  ...    is   conver- 


13.  Show  that  l+-^+T2'"'"T3  ~*"'T4 
gent  for  all  values  of  x 

14.  Show  that  1+T,,    :-7Q"+Tr+  •  •  -^^  convergent  for 

If      If      Ir 
all  values  of  ;p. 


CHAPTER  XXV. 

DNDETERMINBD   COEFFICIBNTS. 

594.  We  know  that =x-\-a,    and   if  we  inte- 

x—a 

gralize  this  we  obtain  an  equation  of  the  second  degree, 
but  an  equation  of  a  different  kind  from  those  treated  in 
chapters  XII  and  XIII,  for  the  equations  previously  treated 
under  the  name  quadratics  were  shown  in  Art.  282  to 
have  two  roots,  and  only  two;  that  is,  it  was  shown  that 
there  were  two  and  only  two  values  of  the  unknown 
number  which  satisfy  the  equation;  but  here  we  have  an 
equation  of  the  second  degree  which  is  satisfied  by  any 
value  whatever  of  x. 

The  reason  is,  that  when  the  equation  is  in  the  integral 
form  we  have  exactly  the  same  function  of  x  on  each  side 
of  the  sign  of  equality. 

595.  Theorem  I.  If  two  functions  of  x  of  the  nth 
degree,  Aq+Aj^x+  •  •  -hA^x""  and Bq+B^x+  •  •  -i-B,,x'\ 
are  equal  for  every  value  of  x,  then  the  coefficients  of  like 
powers  of  X  on  the  two  sides  of  the  sign  of  equality  are  equal 
each  to  each. 

If  the  two  functions  are  equal  for  every  value  of  x,  we 
have 

^0+^1-^+  •  •  •  +^„^=^o+^i-^+  •  •  •  -^B^x^  (1) 
and  since  this  equation  is  true  for  any  value  of  x,  we  may 
consider  ;i:  as  a  variable,  varying  in  any  way  we  please. 

Then  if  we  consider  x  to  approach  a  limit,  each  side  of 
the  equation  is  a  variable  which  approaches  a  limit,  and 
we  have  two  variables  which  are  always  equal,  and  each 


UNDETERMINED    COEFFICIENTS.  425 

approaches  a  limit,  hence  by  Art.  420  the  limits  are  equal. 
Suppose  X  to  approach  zero  as  a  limit,  then  by  Arts.  421 
and  422     limit  of  (^0+^1^+  •  •  •  +^n-^'0=^o 
and  limit  of  (^0+^1-^+  •  •  v  +  ^n^")=^o, 

hence  by  Art.  420  ^  0  =  ^0  •  (2) 

Subtracting  A^  from  the  left  side  and  B^  from  the  right 
side  of  (1)  we  get 

^1^+^2-^24..  .  .J^A^x^^B^x-\-B^x''-\-^  .  •^-B^x^  (3) 
Divide  both  members  of  (3)  by  x  and  we  have 
A^-^A^x-V  •  •  •  ^-A^x^-^^B^^B^x+  .  •    B„x"-^     (4) 

Again  let,^  approach  zero  as  a  limit,  then 

limit -of  (-^1+^2-^+  •  •  •  +A„x"--'^)=A^ 
and  limit  of  CB^  ^B^x^  •  .  •  -\-B„x''-^')==B^, 

therefore,  by  Art.  420,      A^=^B^.  (5) 

Subtracting  A  ^  from  the  left  side  and  B^  from  the  right 
side  of  (4)  we  get 

A^x-^A.^x'^+  .  .  .  +^«^"-i 

=B^x+B^x'^+ |-^«^"-i         (6) 

Divide  both  members  of  (6)  by  x 

^2+^3^+  ■  •  +A„x^-''=B^-^B^x+  .  .  +B„x^-''        (7) 

Then  in  the  same  way  as  in  the  two  preceding  in- 
stances it  follows  that     ^2=^2 
and  by  continuing  the  process  we  get 

^4  =  ^4, 

etc. 
Therefore  if  the  two  functions  are  equal  for  all  values 
of  X,   the  coefficients  of  like  powers  of  x  in  the  two 
functions  are  equal  each  to  each. 

596.  Equations  of  the  kind  just  considered,  which  are 
satisfied  by  any  value  of  x  are  often  called  Identical 
equations,  while  those  which  are  satisfied  by  particular 
values  of  x,  equal  in  number  to  the  degree  of  the  equa- 
tion, are  often  called  Conditional  equations. 


426  UNIVERSITY   ALGEBRA. 

597.  Theorem  II.     The  limit  of  the  sum  of  the  series 

\-\-r-\-r'^  -\-r^-\-  -as  the  7iuniber  of  terms  increases  with- 
out limit  and  as  r  approaches  zero,  is  1. 

In  the  case  of  a  decreasing  geometrical  progression  it 
was  shown  in  Art.  502  that  as  the  number  of  terms 
increases  without  limit 

limit  5=:j 

a 
Jhe  expression  :j will,    of  course,    have   different 

values  for  different  values  of  r^  hence  we  may,  if  we 
choose,  look  upon  this  expression  as  a  variable.     But  as 

r  approaches  zero  as  a  limit,  the  fraction  3— » —  approaches 

QJ  as  a  limit.  Therefore,  we  may  say  that,  in  the  decreasing 
geometrical  progression,  as  the  number  of  terms  increases 
without  limit,  and  the  ratio  approaches  zero  as  a  limit, 
the  sum  approaches  a  as  a,  limit. 

In  particular,  then,  if  a=l  and  the  number  of  terms 
increases  without  limit,  and  the  ratio  approaches  zero  as 
a  limit,  the  series 

1  +  r+r'^  +  r^'i 

approaches  1  as  a  limit. 

598.  Theorem  III       The  limit  of  the  series 

as  the  number  of  terms  increases  without  limit  and  as  x 
approaches  zero,  is  Aq. 

Take  the  series  first  without  A  q  ,  and  suppose  J^  to  be 
a  positive  number,  numerically  equal  to  the  greatest  of 
the  coefficients  A-^,  A 2,  A^,  etc. 

ThenAiX+A2X^+  •  .  •  numerically <A^(jr+jr 2+  .  .  .) 

By  the  preceding  article  the  limit  o{l+x+x'^-{-x^+  -  • 
as  the  number  of  terms  increases  without  limit,  and  x 
approaches  zero,  is  1. 


UNDETERMINED    COEFFICIENTS.  427 

Hence  limit  (x-}-x'^+  -  .  .)=0. 

Hence  limit  J^(x+x'^  -f  •  •  •  )=0. 

That  is  to  say,  the  right  hand  member  of  the  inequality 
above  can  be  made  as  near  zero  as  we  please.  Therefore, 
since  the  left  member  of  the  inequality  is  always  numeri- 
cally less  than  the  right  member,  the  left  member  can 
be  made  as  near  zero  as  we  please. 

Hence      limit  (A^x+A2x'^+A^x^+  .  ^  .)=0. 
Hence  limit  (^0+^1-^+^ 2-^^+  •  •  •)=^o- 

599.  Theorem  IV.  //,  for  every  value  of  x  which  makes 
each  of  the  two  scries  A^-\-A^x-\-  •  -and  Bq+B^x+  •  •  • 
convergent^  these  two  series  approach  the  SAMK  limit  as  the 
number  of  terms  increases  without  limit,  then  the  coefficients 
of  like  powers  of  x  in  the  two  series  are  equal  each  to  each. 

Since  we  are  dealing  with  the  limit  of  a  convergent 
series  as  the  number  of  terms  increases  without  limit,  we 
know  that  by  taking  a  sufficient  number  of  terms  the 
sum  of  the  terms  taken  may  be  made  to  differ  from  the 
limit  of  the  sum  by  an  amount  as  small  as  we  please. 

Let  us  then  write 

limit  {A^^rA^x^-A^x'^^ \.A„_^x*'-^+ >  .  .) 

=Aq-\-A^x+A2X^+  .  .  .  +^«_l^''-l+i?l^^ 
where  ^j^  is  of  course  the  difference  between  the  limit 
of  the  sum  as  the  number  of  terms  increases  without 
limit  and  the  actual  sum  of  the  first  n  terms. 

AQ.A^y'  .  'A„_-^  are  constants,  buti?i  is  not  a  constant, 
for  if  it  were  the  series  would  terminate.  In  fact  jRj^x** 
approaches  zero  as  n  increases,  for  if  it  did  not  the  series 
would  not  be  convergent.  An  inspection  of  the  series 
shows  that  every  term  after  the  first  contains  x,  every 
term  after  the  second  contains  x^^  every  term  after  the 


428  UNIVERSITY   ALGEBRA. 

third  contains  x^,  and  so  on;  hence  every  term  after  the 
nth  will  contain  the  factor  ^",  and  hence  it  is  natural  to 
assume  the  remainder  after  n  terms  are  written  to  be  of 
the/orm  R^x*". 

Instead  of  writing  limit  of  Aq+A^x+  -  •  -as  the  num- 
ber of  terms  increases  without  limit,  we  write 

Aq+A^x-j-A'2X^+  .  .  -A„_-^x''-^+R^x'' 
and  in  the  same  way  we  write 

Bo-hB^x+B^x^+  .  .    ■i-B^.^x^-^+R^x' 
instead  of  writing  limit  of  Bq-\-Bj^x+B2x'^+  .  .  -as  the 
number  of  terms  increases  without  limit. 

Using  this  notation  we  may  write 
Ao+A^x+  .  .  .  -{-A„_^x»-''+R,x^ 

=^0  +^i^+  ■  •  •  +B^_^x^-^  +i?2^.       (1) 

If  now  we  consider  x  as  sl  variable  approaching  zero 
we  have  here  two  variables  which  are  always  equal,  and 
therefore  by  Art.  420,  their  limits  are  equal.  By  Art.  598 
the  limit  of  the  left  hand  member  equals  Aq  and  the  limit 
of  the  right  hand  member  equals  Bq;  hence  Aq=Bq. 

Subtracting  Aq  from  the  left  hand  member  and  Bq  from 
the  right  hand  member  of  (1),  we  get 

A^x-i-A2X^+^  .  .+^1-^" 

=B,x+B^x''+  .  .  ■  +i?2-^         (2) 
Dividing  both  members  of  (2)  by  x,  we  get 

A^+A2X+  .  .  .  +A„_^x^-''-^jR,x''-^ 

==B,  +B^x+  .  .  .  -{-B^x^'-K         (3) 

As  before,  we  have  two  variables  always  equal,  hence 
their  limits  are  equal. 

But  as  X  approaches  zero  the  limit  of  the  left  hand 
member  equals  A  ^  and  the  limit  of  the  right  hand  member 
equals  B^. 

Hence,  by  Art.  420, 

A,=B,. 


UNDETERMINED    COEFFICIENTS.  429 

Repeating  the  reasoning,  we  may  show  successively  that 

-^Z  —  ^Z) 

etc. 

600.  The  theorem  of  the  last  article  enables  us  to 
change  the  form  of  a  function. 

The  method  of  doing  this  consists  in  assuming  a  func- 
tion of  the  required  form  with  unknown  coefficients  and 
then  determining  the  coefficients  so  that  the  function 
assumed  shall  be  identical  with  the  function  proposed. 
The  unknown  coefficients  are  determined  by  placing  the 
proposed  function  equal  to  the  assumed  function,  reducing 
to  the  rational  integral  form,  and  equating  the  coefficients 
of  like  powers  of  the  variable  on  the  two  sides  of  the 
equation. 

If  the  proposed  function  can  be  placed  in  the  assumed 
form  it  will  be  found  that  there  are  as  many  independent 
compatible  equations  as  there  are  unknown  numbers  to 
determine. 

601.  A  function  is  said  to  be  Developed  or  Expanded 

when  it  is  expressed  in  the  form  of  a  series,  the  sum  of 
whose  terms  when  the  number  of  terms  of  the  series  is 
limited,  and  the  limit  of  the  sum  when  the  number  of 
terms  is  unlimited,  equals  the  given  function. 

602.  The  development!  of  functions  is  one  of  the 
most  common  applications  of  the  method  described  in 
Art.  600.  The  process  is  usually  referred  to  as  the 
Method  of  Undetermined  Coefficients. 

We  will  illustrate  the  method  by  working  an  example. 

Let  us  develop  the  fraction  zj— ^ j— 2 ' 


430  UNIVERSITY    ALGEBRA. 

ALSSume 

Multiplying  both  sides  of  (1)  by  1— 3;ir+4;«r2  we  obtain 
2+Zx=A^  +  (Ai-ZA^')x+iA^-ZA^+4:A^-)x^ 

+  (As-SA^+4A,)x»+.  .. 
+  CJ?-SA„^^+4A„_2)x"  +  (-SJi+4A^.^-)x"+^+4J?x"+-'. 

We  see  that  in  the  left  hand  member  the  coefficient  of 
each  power  of  x,  beyond  the  first  power,  is  zero.  Hence, 
equating  coefficients,  we  get 


^0=2. 

^^-3^0=3. 

•.     ^1  =  3+3^0- 

^2-3^1+4^0=0.      . 

,-.     ^2=3^1-4^0, 

^3—3^2+4^1=0.      , 

.-.      ^3=3^2-4^1 

etc. 

etc. 

From  these  equations  the  law  of  the  series  is  so  evident 
that  we  can  write  as  many  more  equations  as  we  please 
without  further  calculation.  From  the  second  column 
of  equations  it  is  evident  that  each  coefficient  after  A  ^ 
equals  three  times  the  preceding  one  minus  four  times 
the  second  preceding  one.  Now  since  Aq=2  and 
^1=3  +  3^0=9,  we  may  substitute  these  values  in  (1) 
and  determine  other  coefficients  by  the  law  just  stated. 

Hence,  we  obtain 

^    ^  +  ^-^^    ^-=2+9;ir+19;i;2+21.y«-13.a:^-123;i;g-317^«... 

As  we  usually  determine  only  a  few  of  the  coefficients, 
and  then  discover  if  we  can  the  law  of  the  series,  so  it  is 
usual  in  the  assumed  series  with  undetermined  coefficients, 
to  write  only  a  few  terms  and  indicate  the  others  includ- 
ing the  remainder  by  dots,  thus : 

^■^^^    .^A^+A^x+A^x^+A^x^+  .  .  . 


Instead  of  using  the  method  of  undetermined  coeffi- 
<iients,  we  might  have  proceeded  by  ordinary  long  division, 
as  follows: 


UNDETERMINED  COEFFICIENTS.  43 1 

2-6x+Sx'^ 
9x-8x'' 
9x—27x'^-}-S6x^ 

19x^~S6x^ 

19x^  —  57x^+76x^ 

21x^--6Sx^^-Mx^ 
-ISx^-Six^ 
-'lSx^-hS9x^-52x^ 


-'12Sx^-{-62x^ 


Here,  as  before,  we  obtain 
2-i-Sx 


2  +  9;r+19;t:2  +  21jr3-13;i:4-123x5  + 


l—Sx-\-4:X^ 

The  advantage  of  the  method  of  undetermined  coeffi- 
cients over  that  of  ordinary  division  consists  in  the  fact 
that,  by  the  method  of  undetermined  coefficients  we  can 
usually,  after  a  few  terms  have  been  determined,  discover 
the  law  of  the  series,  by  means  of  which  as  many  terms 
as  we  please  may  be  written  down. 

EXAMPI^BS. 

Develop  the  following  fractions  by  the  method  of 
undetermined  coefficients  and  also  by  actual  division, 
and  in  each  example  find  the  law  of  the  series. 


1  g_    1+^ 


1+x  "*    l—x+x^ 

1  +  x  \'\-2x-\-Zx'^ 

^'    1-x  ^*    l-2x+Sx^' 

a  _  V-2x-i-Sx^_ 

^'   a^x  '    l-\-2x+Sx^' 

■     2  +  Sx  7-\-bx 


1-^x  '''    Z-^x'''\-2x^ 

\-2x+Zx'^^  ^^'    3-4;r2+2;t:8' 


432  UNIVERSITY   ALGEBRA. 

Compare  the  laws  of  the  series  in  the  developments  of 
the  fractions  in  examples  1  and  4;  also,  compare  the  lawsoi 
the  series  in  examples  5  and  7;  also,  in  examples  9  and  10. 

Query  :  What  controls  the  law  of  the  series  in  the 
development  of  a  fraction? 

Query:  How  does  the  numerator  affect  the  develop- 
ment of  a  fraction  in  the  form  of  a  series  ? 

Query:     What  do  the  results  to  examples  7  and  -8' 
suggest  about  the  development   of  fractions  which  are 
reciprocals  ? 

603. .  It  sometimes  happens  when  we  try  to  develop  a 
fraction  by  the  method  explained,  that  some  of  the  equa- 
tions are  absurd  or  contradict  one  another. 

The  reason  of  this  is  because  the  fraction  cannot  be 
developed  into  a  series  of  the  form  assumed.     Thus  if  we 

try  to  develop ^ 

Multiplying  by  x—x^,  we  obtain 
l==AoX+iA^^A,)x^  +  (A^-A^)x^  +  (A^-A^')x^+.. 
hence  1=0, 

etc. 

But  the  first  of  these  equations  is  false,  hence  we  infer 
that  the  function  cannot  be  developed  into  a  series  of  the 
assumed  form. 

But  we  note  the  denominator  of  the  given  fraction  con- 
tains a  factor  .;r,  and  that  hence  the  fraction  proposed  equals 

—  X  :j J  the  second  factor  of  which  is  easily  developed, 

giving      -——-  =  1  +X'^x'^-\-x^+x^+  •  •  • 


UNDETERMINED  COEFFICIENTS.  433 

^rom  this  we  infer  that  the  development  of ^  can 

be  obtained  from  the  development  of  z by  dividing 

every  term  in  that  development  by  x. 

Hence         —-^=x-^  +  l+x+x^+x^+  •  .  . 
and  we  would  obtain  this  very  result  if  should  assume 

2  ^^^  0"^  +-«4i 'T-^2-^~r'^3'^  ~r  •  '  • 
or  in  other  words  if  we  should  degin  the  assumed  series 
with  a  term  containing  x~'^  instead  of  beginning  with  an 
absolute  term.  If  the  fraction  we  wish  to  develop  is  in 
its  lowest  terms,  and  if  the  lowest  power  of  x  that  appears 
in  the  denominator  is  the  rth  power,  then  we  must  begin 
our  assumed  series  with  a  term  containing  x'^. 

This  is  a  safe  rule  whether  the  fraction  is  in  its  lowest 
terms  or  not,  but  it  is  not  always  necessary  when  the 
fraction  is  not  in  its  lowest  terms. 

In  any  case,  when  we  form  an  equation  by  putting  a 
given  fraction  on  the  left  and  an  assumed  series  on  the 
right  side  of  the  sign  of  equality,  the  assumed  series  must 
begin  with  such  a  power  of  x  that  when  the  equation  is 
integralized  the  lowest  power  of  x  on  the  right  side  of  the 
equation  will  be  as  low  as  the  lowest  power  on  the  left 
side. 

ISXA.MPI,i;S. 

,.Bevo>op^!.  ^  Develop  il=^. 

604.  Not  only  fractions  but  some  irrational  expres- 
sions may  be  developed  by  the  method  of  undetermined 
coefficients. 

28  — U.  A. 


434  UNIVERSITY  ALGEBRA. 


I^t  US  develop  Vl—x. 

Assume 

Squaring  each  side,  we  get 
l—x=-Ao^  +  2AoA^x+(2A^A^+Ai^)x' 
+  (2A^At  +  2AiA^-)x'^  +  (2A^A^+2AiA^+A,'')x* 
+  (2AoA,  +  2A^A^+2A^As)x^+  .  ■ 
Equating  coefficients  of  like  powers  of  x,  we  get 
^0  =  1. 
2A,A,  =  -1, 
2A^A^+A^^=0, 

2^0^4  +  2^1-48+^2^=0, 

2^0^5  +  2^1^4+2^2^8=0, 

2A,A,+2A^A^+2A^Ai+Ai^'='0, 

etc. 
From  these  we  get 
A,=l, 


A,  =  - 
A=- 


2^0 
A,^ 


^8=- 


A^=- 


A,=- 


2Ao 
2A,A^ 
'    2A,   ' 
2A^A^+Aj^ 
2A,         ' 
2AiAi  +  2A,Ag 


2A, 


A,  =  - 


2A,As  +  2A^A^+At* 


2^0 


etc. 
From  these  the  law  of  the  series  can  be  seen. 


UNDETERMINED  COEFFICIENTS.  435 

Taking  these  equations  in  order,  we  find  the  numerical 
values  of  the  undetermined  coefficients  to  be  as  follows  : 

A 7  A 21 

^6 —       ^S'B"?       -^6—        1024* 

Making  these  substitutions  in  the  assumed  development, 

we  obtain 

X    x^     x^     5x^  '  7x^     21;r« 


^       "^^       2""'8"""l6~128"~256""i024" 


KXAMPI^KS. 


VX^                                       /  1         1         \T- 
l  +  1x—-^'    4.     Develop   (-gH h^j 

2.     Develop  l/^T^-  5-     Develop   (1+^x'')^ 


3.     Develop  (1  +  ^)^.  6.     Develop  ;t:(l+:ir+ ;i;2)ir- 

605.  It  is  interesting  to  note  that  the  development  of 
an  irrational  expression  may  turn  out  to  be  a  series  of  a 
limited  number  of  terms. 

Suppose,  for  example,  we  wish  to  develop  Vl—2x+x^ 
and  do  not  recognize  that  l—2x+x'^  is  a  perfect  square, 
then  assume  as  before 


V1—2x-^x^=Aq+A^x+A^x'^+^  .  . 
Squaring  both  sides,  we  obtain 
l-'lx+x^=^A^^  +  2A^A^x+(,A,''  +  2A^A^)x^ 

+  (2AoA^+2A,A^')x^. 
Equating  coefficients  of  like  powers  of  x,  we  get 

Aq  =1,  .*.        AQ    =    ly 

2AoA, 2.'.A^  =  -1, 

^1^+2^0-42  =  1  •••^2=0, 
2A^A^+2A^A^=0.-.A^=0, 
Ai-'+2A^A^+2A,A^=0  .:  A^^O, 
etc.,  etc., 


436  UNIVERSITY   ALGEBRA. 

and  each,  of  the  subsequent  coeflScients  will  turn  out  to  be 
zero,  hence  we  get 

606.  In  developing  irrational  expressions  it  sometimes 
happens  that  we  should  deg-zn  our  assumed  developments 
with  some  negative  power  of  x. 

An  inspection  of  the  proposed  example  will  show  with 
what  power  of  x  the  development  should  begin ;  for  the 
assumed  series  must  be  such  that,  when  the  equation 
obtained  by  putting  the  given  function  equal  to  the 
assumed  series  is  reduced  to  the  rational  integral  form, 
then  the  lowest  power  of  x  on  the  side  which  contains 
the  undetermined  coefficients  must  be  as  low  as  the  lowest 
power  on  the  other  side  of  the  equation. 

Thus,  to  develop -i/lH — ^   we    begin    the    assumed 

series  with  a  term  containing  ^"^ ,  for  when  this  is  squared 
the  lowest  power  of  x  is  x~^  and  when  both  sides  are 
multiplied  by  x^  to  reduce  to  the  integral  form  then  the 
series  on  the  right  side  of  the  equation  will  begin  with  an 
absolute  term  as  it  should. 

607.  If  we  wish  to  develop  the  algebraic  sum  of  two 
or  more  radicals  it  is  best  to  develop  each  one  by  itself 

and  then  find  the  algebraic  sum  of  the  results. 

KXAMPI^KS. 


I.  Develop  yi~-2. 


^^+j_+T/r 


2.  Develop -i/^+-2  + 1/1+^^. 


3.  Develop  Vl+4x+6x^  +  6x^+5x^  +  2x^+x\ 


UNDETERMINED  COEFFICIENTS.  43/ 


5.  Develop  ^^-^)^~^,. 

6.  Develop  (;«r2+Ar-'2)T+(2;tr2+:r-2)T 


7.  Develop  (l+;r)2+jq- 

_1 

8.  Develop  i 

•^■^ — 2" 


X 


9.   Develop  1/1  + 2jr+3;r2+2:r3+jt-*. 
10.  Develop  (^r^— 4^7+4^6+e^4_i2^3+9)| 


CHAPTER  XXVI. 

SUMMATION   OF  SKRIKS. 

608.  No  general  method  can  be  given  for  finding  the 
sum  of  series  which  will  be  applicable  to  series  of  all 
kinds,  but  in  certain  cases  the  sum  may  be  found  and  it 
is  the  object  of  this  chapter  to  explain  how  to  proceed  in 
some  of  these  cases. 

SKRIKS  REDUCIBI^B  TO  THE)  FORM 

609.  In  Art.  563  it  was  shown  that  an  infinite  series 

is  convergent  when  it  is  capable  of  being  expressed  in 

the  form 

u^'-U2-\-U2—u^+Uq—u^+  ...  (1) 

provided  only  that  limit  tc„=0  as  n  increases  indefinitely. 

But  since  the  sum  of  the  series  (1)  is  u^  it  follows  that 
we  can  find  the  sum  of  any  infinite  series  which  is 
reducible  to  this  form. 

For  example,  the  series 

'        ■  '  +^^^+.--       (2) 


xiix+l)  '  (x+lXx+2)  ■  (^+2)(^+3) 
may  be  expresed  in  the  form  (1) 

The  nth  term  of  (2)  is 


lx+(n~l)']\_x-hn'] 
which  is  easily  seen  to  be  equal  to 

1 1__ 

x+(n—l)      x+n 


SUMMATION   OF   SERIES.  439 

Now,  as  the  series  (2)  is  obtained  by  taking  all  the 
terms    obtained    by    giving    to    n    in    the    expression 

.= — -7 3-r=rT — ; — ^  all  positive  integral  values,  it  follows 

that  a  series  equivalent  to  (2)  may  be  obtained  by  giving 

.      .                   .               1  1        ,,        .  .      . 

n\VL  the  expression  —— r^ -— -  all  positive  integral 

values. 
Hence,  (2)  may  be  written 

By  the  form  of  (3)  it  is  evident  that  the  sum  is  — >  and 

1  "^ 

hence  the  sum  of  the  series  (2)  is  also  — 

X 

610.  It  is  well  to  notice  that  the  sum  of  a  limited 
number  of  terms  of  a  series  reducible  to  the  form  (1)  may 
be  easily  obtained. 

For  example,  the  sum  n  terms  of  the  series  (2)  is  evi- 
dently the  sum  oil  n  terms  the  series  (3);  /.  <?.,  this  sum 
equals 


X     x-hl     x+1     x+2  x+n—1     x+n 

and,  as  all  the  terms  cancel  except  the  first  and  last, 
therefore  the  sum  of  2  n  terms  of  (3)  equals 

1 1    ^       n 

X    x+n    x{x+n) 

EXAMPIvKS. 

Find  the  sum  of  the  following  infinite  series;  also,  the 
sum  of  the  first  n  terms: 
111 
^'   1.2"^2.3"^3.4"^*  '* 

ct a  A.  ^  _L. 

x{x-\-aj     (x+a)(x-\-2a)     (x-{-2a)(x-{-Sa) 


440  UNIVERSITY  ALGEBRA. 

+  0 — TT^i    •  •  • 


^*   2.4^4.6  '  6.8  '  8.  10 

2  11 

Here  the  nth  term  equals  2n2{n  +  l)'  "^^^^^  ®^"^^^  2^''2^+T) 

2  2  2 

2 

Here   the    nth  term   equals  /2n  +  l)i2n+dY   ^^^^^  plainly  equals 
1  1 


2»+l       2n-\-d' 


22_i     32—22     42—32 


^*     1  .  2    '  22  .  32  '  32  .  42  ' 

SUMMATION  BY  UND^TKRMINKD  COEFFICIENTS. 

611.  Sometimes  the  sum  of  a  finite  series  may  be 
obtained  by  the  method  of  undetermined  coeflScients. 
This  will  usually  happen  when  the  sum  of  the  n  terms 
is  equal  to  an  expression  containing  n,  which  expression 
has  the  same  form  no  matter  what  value  n  has. 

To  illustrate  this,  let  us  take  the  series 
12+22+32+42+.  .  .+«^ 

Assume  12 +22 +32 +42+  .  .  .  +;^2 

=^+i9;^+0^2+Z?;^8  4..  .  ^  (y^ 

If  this  equation  is  to  remain  true  whatever  be  the  value 
of  n^  we  may  change  n  into  n+X, 

Making  this  change,  we  obtain 
12+22  +  32+42+  .  .  .  +(;^  +  l)2 

=^+^(^  +  l)  +  C(^  +  l)2+Z7(;2  +  l)3+  ...       (2) 

Subtracting  (1)  from  (2),  we  obtain 
(7^+1)2=^+ C(27^+l)  +  ZP(3;^2+3;^  +  l) 

+^(4;^3+6;^2+4;^  +  l)+.  .  . 


SUMMATION    OF    SERIES.  44I 

This  equation  may  evidently  be  satisfied  by  making  E 
and  each  of  the  subsequent  coefficients  equal  to  zero  and 
equating  the  remaining  coefficients  of  like  powers  of  n 
on  the  two  sides  of  the  sign  of  equality. 
We  thus  have 

3Z>=1  .-.  D=\, 

3Z>+2C=2        .-.   C=^. 
D^C^-B=1      .-.  ^=|. 
Substituting  the  values  here  found  in  equation  (1),  we 
have  12+22+32+42=^+1+^+^. 

Since  this  equation  is  to  hold  for. all  values  of  n,  we 
may  determine  A  by  assigning  to  7i  any  value  we  please. 
Making  ;?=!,  we  obtain 

Therefore  ^4=0,  and  we  get  as  our  final  result 

2 


n     n"  ^  n 


12+22+32+   . .  +^2=-+-_+:__ 

^n^ -^Zn'' -{-n  _n{n  +  \X^n  +  r) 
6       "     ■"  6 

KXA.MPLKS. 

1.  Show  that  12  +  32+52+.  .  .+(2;^— 1)2 

3 

2.  Show  that  22+42+62+ ... +(2;^)2 

_2;^(7^  +  l)(2;^  +  l) 
3 

3.  Show  that  18+23+33+ |-;^3^^y(^  +  l)l^ 

4.  Show  that  13  +  33  +  53  +  . ..  +  (2;2—l)3=^2(2^2__x) 

5.  Show  that  23 +43 +  63+.  .  . +(2;^)3  =  2;^2(^  +  l)2^ 

Show  that  1.2  +  3.4  +  5.6+  .  .  . +(2;^— 1)(2;^) 
_;^(;^  +  l)(2;^~l) 

3 


6 


442  UNIVERSITY    ALGEBRA. 

METHOD   OF   DIFFKRKNCES. 

612.  If,  in  any  series,  each  term  be  subtracted  from 
the  succeeding  term,  the  various  remainders  form  a  new- 
series  called  the  First  Order  of  Differences  of  the 
given  series. 

If,  in  this  new  series,  each  term  be  subtracted  from  the 
succeeding  term,  the  various  remainders  form  still  another 
series  called  the  Second  Order  of  Differences  of  the 
given  series. 

In  the  same  manner  from  this  last  series  a  new  series 
may  be  formed  called  the  Third  Order  of  Differences 
of  the  given  series,  and  so  on. 

For  example,  if  the  given  series  be 

6,  11,  23,  45,  80,  131,  201, 
we  have  the  following  series : 

Given  series:  6,  11,  23,  45,  80,  131,  201; 

1st  order  of  differences:     5,  12,  22,  35,  51,   70; 
2d  order  of  differences  :(:■     7,  10,  13,  16,    19; 
3d  order  of  differences:         3,     3,    3,     3; 
4th  order  of  differences :  0,     0,     0. 

613.  The  method  of  finding  any  term  of  a  given 
series,  or  the  sum  of  any  number  of  terms,  by  means  of 
its  successive  orders  of  differences,  is  called  the  Method 
of  Differences. 

614.  To  find  any  term  of  a  given  series, 

Let  the  terms  of  the  given  series  be  represented  by 
^1,  z^2>  ^3'  ^4.  etc. 

Forming  the  successive  orders  of  differences,  we  have 
1st   order  z^ 2 — ^u  ^3 — ^2j  ^4"~^3>  ^5 — ^4*  etc. 
2d    order  2^8—22^2  +  ^^1'  u^  —  ^u^-^-Uc^,  u^—2u^+u^,  etc. 
3d    order  2^4  — 3^3  +  02^2  — z^i,  2^5— 3^^4^-3^^3— z^2>  etc. 
4th  order  u^— 4:21^ -\-6u.^—4iU 2  + u^y  etc. 


SUMMATION   OF   SERIES.  445 

Let  the  first  terms  of  these  successive  orders  of  differ- 
ences be  represented  by  d^,  d^,  d^,  d^,  etc.,  then 
d^  =  U2 — ^1. 

(^2  =  ^3 — 2z^2+^l- 

=  2^1+2^1+^2- 

.*.   2^4  =  2^1— 32^2 +^^^3 +^3- 

=  2^1-3(^^1 +^l)  +  3(ZiJl +2^1 +<)+^3. 
=xZ^l+3^1+3^2+^3. 

We  notice  in  the  value  of  2^3  the  numerical  coefficients 
are  the  same  as  in  the  expansion  of  (a  +  xy  and  in  the 
value  of  the  u^^  the  numerical  coefficients  are  the  same  as 
in  the  expansion  of  (a+;r)^,  and  from  this  we  are  led  to 
think  that  probably  the  same  law  holds  good  generally  ; 
that  is,  that 

^(.:.l)(.-2)(.-3)^^_^    _^     (1) 

615.  We  will  now  prove  by  mathematical  induction 
that  this  law  is  true  generally. 

Let  us'  assume  for  the  moment  that  this  law  holds  for 
the  ^th  term  of  any  series,  then  it  must  hold  for  the  first 
order  of  differences  of  that  series.  Writing  down  the  :^th 
term  of  the  first  order  of  differences  by  this  law,  remem- 
bering that  the  nth  term  equals  u„+^—u„,  and  that  d^j 
^2,  <^3,  etc.,  for  the  first  order  of  differences  correspond  to 
u^y  d^,  d^,  etc.,  for  the  given  series,  we  have 

I  {n-l){n-2-)(n-^) 

T IQ "4+       •• 


444  UNIVERSITY   ALGEBRA. 

Adding  this  equation  to  the  preceding,  we  obtain 

,  fin-lXn-2Xn-3)  ,  (n-iyn-2)\_^ 
+[  [3  +  [2  /»+••• 

Hence,  «^i=»«i+«^iH — ^-r^-d^ 

^<n-lXn-2)^^_^     ^^        (2) 


|3 

This  expression  for  u^^-^^  is  in  accordance  with  the 
above  law.  Therefore,  zf  the  law  holds  for  the  T^th 
term  of  any  series  it  also  holds  for  the  (;^^-l)st  term 
of  that  series.  But  we  know  that  the  law  holds  for  the 
fourth  term  of  any  series;  therefore,  it  holds  for  the  fifth 
term,  and  holding  for  the  fifth  term  it  holds  for  the  sixth 
term,  and  so  on.     Hence,  the  law  is  general. 

616.     To  find  the  sum  of  any  number  of  terms  of  a 

series. 

Let  the  terms  of  the  series  be 

e^i,  u^,  «3,  u^,  etc., 
and  let  Sn  represent  the  sum  of  the  first  n  terms  of  this 
series. 

Let  us  form  the  series  of  which  the  given  series  is  the 
first  order  of  difierences.     This  series  is 

0,   ^1,   «2+^l'   ^3+^2+^<^i,   2^4+^3+^2+^1)   etc. 

Evidently  now  the  (;^  +  l)st  term  of  this  series  is  equal 
to  the  sum  of  the  first  n  terms  of  the  given  series,  and  our 
problem  reduces  to  that  of  finding  the  (7^4-l)st  term  of 
the  series  just  written. 

To  find  this  term,  we  use  the  formula  already  found, 
viz:  (1)  Art.  614,  substituting  kS^  for  u^,  («+l)  for  », 
0  for  ^1,  z^i  for  ^i,  d^  for  d^.  etc.     Hence, 

^      /%  .           .  n{n—V)  -    ,  n(n—V){n—^  .    , 
S„^0  +  nu^+    ^  .^    V,+    ^        .^^ ^^3+  .  .  . 


SUMMATION   OF   SERIES.  445 

That  this  method  may  be  applied  to  the  sum  of  a  series 
it  is  necessary  that  when  successive  orders  of  differences 
are  formed  one  is  finally  reached  whose  terms  are  all  zeros, 
and  in  this  case  the  expression  above  for  S„  will  be  the 
sum  of  a  limited  number  of  terms. 

KXAMPLKS. 

1.  Find  the  tenth  term  of  the  series 

l  +  3,+6,  +  10,  +  15,.... 

2.  Find  the  nth  term  in  the  series  in  example  1. 

3.  Find  the  sum  of  10  terms  of  the  series  in  example  1 . 

4.  Find  the  sum  ofn  terms  of  the  series  in  example  1. 

5.  Find  the  ^th  term  of  the  series 

2+4+9+16  +  25+..  . 

6.  Find  the  sum  of  n  terms  of  the  series  in  example  5. 

7.  Find  the  nth  term  of  the  series 

1+5  +  12+22  +  35+ .  .. 

8.  Find  the  sum  of  7  terms  of  the  series  in  example  7. 

9.  Find  the  sum  of  8  terms  of  the  series 

1+6  +  15+28+ .  .  . 
iO.  Find  the  sum  of  8  terms  of  the  series 
4+9+18+31+ .  .. 

r:^curring  series. 

617.     In  chapter  XXV  we  found  (that 
2+3;c 


=2  +  9ji:+19^'^  +  21;tr8-13;c4_i23^5. 


1— 3jr+4;t:2 

In  the  series  on  the  right  hand  side  of  this  equation, 
each  term  after  the  second  term  is  equal  to  Sx  times  the 
preceding  term  minus  Ax^  times  the  second  preceding 
term.  This  illustration  will  enable  us  to  understand  the 
following  definition. 


44^  univp:rsity  algebra. 

618.  A  Recurring  Series  is  one  in  which,  after  some 
particular  term,  each  subsequent  term  is  equal  to  the  algebraic 
sum  of  a  certain  {number  of  preceding  terms  multiplied  by 
certain  numbers  which  are  the  same  zvhatever  term  of  the 
series  is  being  found. 

619.  A  recurring  series  in  which  each  term  depends 
upon  only  one  preceding  term  is  said  to  be  of  the  First 
Order ;  one  in  which  each  term  depends  upon  two  pre- 
ceding terms  is  said  to  be  of  the  Second  Order ;  one  in 
which  each  term  depends  upon  r  preceding  terms  is  said 
to  be  of  the  rth  Order. 

620.  The  equation  which  expresses  any  term  in  a 
recurring  series  of  the  rth  order  in  terms  of  the  r  pre- 
ceding terms  is  called  the  Identical  Relation,  and  when 
all  the  terms  of  this  equation  are  transposed  to  the  left 
member  the  sum  of  the  numbers  by  which  the  terms  of 
the  series  are  multiplied  is  called  the  Scale  of  Relation. 
Thus,  in  the  above  illustration,  the  identical  relation  is 

or  —\Zx'^=^?>x.1\x^—^x'^.Vdx'^, 
or  -123;tr5  =  3^(-13;f4)-4;tr2.21;r8, 
or  any  other  equation  obtained  from  the  given  series  by 
the  same  law  as  that  used  in  obtaining  the  equations  here 
written.  Of  course  if,  in  any  of  these  equations,  all  the 
terms  were  transposed  to  one  member  the  resulting  equa- 
tion would  be  called  the  identical  relation.  If  all  the 
terms  of  the  identical  relation  here  written  be  transposed 
to  the  left  member  and  if  we  then  write  the  sum  of  the 
coefl5cients  of  those  terms  of  the  series,  which  appear  in 

the  identical  relation,  we  obtain  the  scale  of  relation 
1—3.^+4^2 

no  matter  which  of  the  above  equations  is  taken  for  the 
identical  relation. 


SUMMATION   OF   SERIES.  447 

621.      To  find  the  scale  of  relation  of  a  recurring  series. 

If  the  series  is  of  the  first  order  its  scale  of  relation 
contains  two  terms ;  if  of  the  second  order  the  scale  of 
relation  contains  three  terms ;  if  of  the  ;'th  order  its  scale 
of  relation  contains  r-\- 1  terms.  The  identical  relation 
of  course  contains  the  same  number  of  terms  as  the  scale 
of  relation. 

Let  the  terms  of  the  series  considered  be  represented 
t>y  ^i>  ^2)  ^3'  ^t^M  ^^^  l^t  ^^  suppose  first  that  the  series 
is  of  the  first  order,  then  the  identical  relation  is  plainly 

2/«+M.-i=0  (1) 

where  p  is  some  coefficient  at  present  unknown. 
The  scale  of  relation  in  this  case  is,  by  definition, 

zc 
From  equation  (1)  it  follows  th.2Ltp= —  and  hence 

the  scale  of  relation  is     1  - 


Next,  let  us  suppose  the  series  is  of  the  second  order, 
then  the  identical  relation  is 

u„-j-pu„_^+gu„_2=0      '  (2) 

and  the  scale  of  relation  in  this  case  is,  by  definition, 
1+p  +  q. 
From  the  identical  relation  we  have 

u„-^pu„^^-\-gu„_2=^y  •    (2) 

and  u„^  1  +pu,,  -f  qti„_  ^  =  0.  (3) 

Equations  (2)  and  (3)  are  sufficient  to  enable  us  to  find 
values  ofp  and  g  and  when  found  the  scale  of  relation  is 
obtained  by  substituting  these  values  in  the  expression 
1+p-^g. 
Similarly,  if  the  series  is  of  the  third  order  we  have 
three  equations  from  the  identical  relation,  viz: 
u„+pu„_j^  +gu„_2+ru^_j^=0 

««+ 1  -^P^n + qun^  J  +  ru^-  2 = 0 
««+ 2 +j^^«+ 1  +  ^«« + ^/^«- 1  =  0 


44^  UNIVERSITY   ALGEBRA. 

and  from  these  three  equations  we  can  find  the  values  of 
p,  q,  r,  which  substituted  in  the  expression 

give  the  scale  of  relation. 

In  a  similar  manner,  if  the  series  is  of  the  rth  order, 
the  scale  of  relation  may  be  written  down  containing  r 
unknown  numbers  and  the  identical  relation  gives  us  r 
equations  from  which  to  determine  these  f  unknown 
numbers. 

622.  If  a  series  is  known  to  be  recurring  but  its  order 
is  unknown  we  first  make  trial  of  a  scale  \-\-p  and  by- 
applying  this  scale  to  two  consecutive  terms  of  the  series 
we  write  down  the  identical  relation  and  then  applying 
the  scale  to  two  other  consecutive  terms  we  write  down 
the  identical  relation  again,  and  so  on  as  many  times  as 
we  please.  From  any  one  of  the  identical  relations  thus 
written  we  determine  the  value  of  p  and  if  the  value  thus 
found  satisfies  all  the  other  equations  written,  the  true 
scale  of  relation  is  obtained  by  writing  this  value  in  place 
of  p  in  the  expression  1  +/. 

If  the  value  of  p  determined  as  just  explained  does  not 
satisfy  the  other  equations  we  have  not  found  the  true 
scale  of  relation  of  the  series  and  we  assume  another 
scale,  viz:  1+^^+^,  and  applying  this  to  three  consecu- 
tive terms  of  the  series  we  write  the  identical  relation, 
and  then  applying  the  scale  to  three  other  consecutive 
terms  of  the  series  we  write  the  identical  relation  again, 
and  so  on  as  many  times  as  we  please.  From  any  two  of 
these  equations  we  may  determine  p  and  q,  and  if  the 
values  thus  found  satisfy  all  the  other  equations  written, 
the  true  scale  of  relation  is  obtained  by  substituting  the 
values  of;i>  and  q  thus  found  in  the  expression  \'\-p-\-q. 

If  the  values  of  p  and  q  found  as  just  explained  do  not 


SUMMATION   OF   SERIES.  449 

satisfy  all  the  subsequent  equations  we  have  not  yet 
found  the  scale  of  relation  of  the  series  and  must  assume 
another  scale  l+p-\-q+r  and  proceed  in  a  similar  man- 
ner and  so  on  until  finally  a  scale  is  found  which  leads  to 
no  inconsistencies. 

KXAMPi^KS. 

1.  Find  the  scale  of  relation  of  the  series 

Assuming  the  scale  of  relation  of  the  series  to  be  1-1-/,  we  have 

From  the  first  of  these  equations  /=  — _,  but  this  value  of  p  does 

2 
not  satisfy  either  of  the  other  equations.     Hence  the  scale  of  relation 
is  not  of  the  form  1  -f /. 

Assuming  the  scale  of  relation  to  be  of  the  form  1  -\-p  -\-  q,  we  have 

— 3a;5  ^px^  +  2qx^  =0 
2x6— 3/;«:5  +  ^^4— 0 

%7  +  2/^6-3^^s=0. 
From  the  first  two  of  these  equations  we  find  p^=x,  q—x^,  and  as 
these  values  of  /  and  q  satisfy  the  other  two  equations  we  conclude 
that  1  +  ;*: 4-^:2  is  the  scale  of  relation  of  the  given  series. 

2.  Find  the  scale  of  relation  of  the  series 

x—x^-i-x^—x'^-{-x^—x^^-\-  .  .  . 

3.  Prove  that  2+4x+14:X^  +  A6x^  +  152x^+  ...  is  a 
recurring  series  of  the  second  order. 

4.  Prove  that 

l  +  6x-lbx^  +  57^'-159x^  +  48dx^-'U55x^+  .  .  . 
is  a  recurring  series  of  the  second  order. 

623.      To  find  the  sum  of  a  Recurring  Series. 

The  method  of  finding  the  sum  of  a  recurring  series  will 
be  clear  if  we  give  the  explanation  for  a  series  of  the 
second  order. 

29  — u.  A. 


450  UNIVERSITY   ALGEBRA. 

Let  Ui,  «2»  ^3'  ^4'  ^^c.,  represent  the  terms  of  the 
series  and  let  Sn  represent  the  sum  of  the  first  n  terms, 
and  if  the  series  be  an  infinite  converging  series,  let  kS 
represent  the  sum  of  the  series,  and  let  1+p+gr  represent 
the  scale  of  relation,  then  we  have 

pSn==      pu^ ■\-pu^ -\-pu.^-^  .  .  .  +pu„_^ ^-pUn 

+  (u„-\-pUn-^  ■\-qUn-.'2)  +  {^pUn-\-qUn-^  +  qu^. 

As  the  terms  here  grouped  (each  expression  in  a  paren- 
thesis being  considered  as  one  term)  it  is  evident  that  the 
identical  relation  makes  each  term  of  the  second  member 
except  the  first  two  and  the  last  two,  equal  to  zero.   Hence, 

Sn-\-pSn+qSn=u^+{u^+pUy)^(^pu,,+qu„_^)-{-qu„ 
.      ^  _(ui+u^+pu^)  +  {pu„+qu„_^-{-qu,;) 
1+P^-q 
If  the  series  proposed  is  an  infinite  convergent  series 
the  expression  {pu,,-{-qu„_^+qu^  approaches  the  limit 
zero  as  n  increases  indefinitely,  and  therefore  for  an  infinite 
converging  series,  we  have 

u^+u^+pu^ 
1+p+q 

624.  In  case  of  a  recurring  series  arranged  according 
to  increasing  powers  of  x^  as  for  example  the  series 
a^+a^x+a<2,x'^-\-a^x^+  •  •  .,  the  two  formulas  of  the 
last  article  become 

_(aQ-^a^x+paQx)-\-x''{pan_^+qa„^o,-^qan-iX) 
^'^  1+px+qx'' 

a^+a^x+pa^x 
l^px-\-qx^ 


SUMMATION   OF   SERIES.  45 1 

Now  it  must  be  remembered  that  the  expression 

1+px+gx^ 
is  the  sum  of  the  series 

a^-j-a^x+a^x^ -\-a^x^  +  .  .  . 
when  and  only  when  this  series  is  convergent. 
But  if  the  fraction 

aQ-\-a^x-{-paQX 
1+px+qx'^ 
be  developed  according  to  ascending  powers  of  x  that 
development  will  be  the  series 

aQ—a-^x+a2x'^-\-a^x^+  •  •  . 
whether  this  series  is  convergent  or  not.     For  this  reason 
the  fraction 

aQ  +  a^x-\-paQX 
l-j-px+qx^ 
is  called  the  Generating  Function  of  the  series 
aQ+a^x-\-a2x'^-^a^x^+  •  .  • 
Thus  we  see  that  the  generating  function  of  a  recur- 
ring series  is  the  sum  of  the  series  only  when  the  series 
is  convergent. 

KXAMPI^ES. 

1.  Find  the  sum  of  six  terms  of  the  series 

l+2x+Sx'^+4:X^+bx^+.  .  . 

2.  Find  the  generating  function  of  the  series  in  ex- 
ample 1. 

3.  Find  the  generating  function  of  the  series 

l+x^-x^^^+2x^Sx^-i-5x^Sx'^+  .  .  . 

4.  Find  the  generating  function  of  the  series 

5.  Find  the  generating  function  of  the  series 

l  +  2x^-2x'^+4x^-6x^  +  10x^—16x'^-i-  •  .  . 


452  UNIVERSITY   ALGEBRA. 

6.     Find  the  generating  function  of  the  series 

.     7.     Find  the  generating  function  of  the  series 

8.  Find  the  generating  function  of  the  series 

i  +  i^-|^2+|3^3_2  9^4+   .    .    . 

9.  Find  the  generating  function  of  the  series 

10.     Find  the  generating  function  of  the  series 
l+x+2x^  +^x^  +bx^  +  ^x^  +  .  .  . 


CHAPTER  XXVII. 

BINOMIAI.  THEOREM  FOR  FRACTIONAI.  AND  NEGATIVE 
EXPONENTS. 

625.  In  chapter  XXII  it  was  shown  that  when  n  is  any 

positive  integer  we  have 

n{7i  —  V)    „  .  n(n — ])(^^— -2)    _  , 
(l+;^)«=l+;^Jr+-^-|^^;^:2  +  -^ )^ L^z  j^  .  .  . 

We  shall  prove  in  this  chapter  that  the  same  equation 
holds  for  certain  values  of  x  when  n  is  any  negative 
integer  or  any  positive  or  negative  fraction. 

626.  Exponent  a  Negative  Integer.  By  division, 
we  get 


1+x  1+x 

If  X  is  intermediate  between  +1  and  —1,  that  is,  if 
l>;r>  —  1,  then  as  n  increases  indefinitely  the  remainder 
approaches  zero,  and  we  may  write 

j^-=l-x+x^-x^+....  (1) 

If  !>;»;>  — 1  this  series  is  absolutely  convergent.  We 
may,  therefore,  by  Art.  593,  raise  both  members  to  any 
power  by  multiplication  without  liability  of  error.  We 
get 

=  l—2x-\-Sx'^'-4:X^+  ...         (2) 
=  l~3;r+6;i:2_  10:^3+  ...  (3) 


454  UNIVERSITY   ALGEBRA. 

The  products  (2)  and  (3)  may  be  expressed  as  follows: 

-2(-2-l>2 


(l^x)-^  =  l  +  C-2)x- 


L2 

2— nr — 9— 9> 


-2(-2-l)(-2-2) 


_^-3(-3-g(-3-2)^3^^^^         (4) 

We  see  from  this  that  when  the  exponent  ;e=  —  2  or  —3 
the  law  of  coefficients  is  the  same  as  when  n  is  sl  positive 
integer.  Does  this  law  hold  for  all  negative  integral 
values  of  the  exponent  ? 

Suppose  it  holds  n=—r,  then  we  have 

(l+^)-''=l+/i^+/>2-^'+  •  •  •  +/X+  •  •  •       (5) 

where  /i  =  —  ^,  ^2  = To '  ^^^ ^^  ^^*   Multiply  the 

[^ 
members  of  equation  (5)  by  the  members  of  (1),  and  we 

get 

+  (A-A-i+-  ..+1K+...      (6) 

But    A-1^ r^U,    A"-/i  +  l=^~'^""^|2""''~' 

Thus  the  coefficients  of  x  and  ^ir^  in  (6)  are  seen  to  be 
the  regular  binomial  coefficients.  Is  this  true  of  all  the 
coefficients  in  (6)?  Suppose  the  law  is  true  for  the 
coefficient  of  x^~'^.  Then  that  coefficient  will  be 
(-r~l)(~r~2).  .  .(~r-^+l) 


^-1 


Observe  that  the  co- 


efficient of  x^  in  (5),  minus  the  coefficient  of  x^  ^  in  (6), 


BINOMIAL   THEOREM.    ANY   EXPONENT.         455 

equals  the  coeflScient  of  x"  in  (6).     Hence  the  coefficient 

of  x'  in  (jo)  is  equal  to 

-^(_^_l)..(-^-5+l)     (-r-lX-r-2)..(-r-s+l-) 


s-1 


(-r-l)(-r-2)..(-r-^) 

= j > 

s 
which  is  the  regular  form  for  the  coefficient  of  x^  in  the 
binomial  formula.  Hence,  if  the  coefficient  of  x^~'^  obeys 
the  law,  then  does  also  that  of  x^.  But  the  coefficient  of 
x^  does  it;  therefore,  that  of  x^  does  it,  and  so  on.  We 
see,  therefore,  that  the  law  of  coefficients  holds  for  all 
the  terms  in  (5),  and,  therefore,  holds  when  the  expo- 
nent n= — r—l  if  it  holds  for  n= — r.  But  it  has  been 
shown  above  to  hold  for  n=  —  S;  therefore,  it  holds  for 
n^—4,  and  so  on,  for  any  negative  exponent.  Thus,  we 
have  found  the  formula 

(l+xr=l  +  nx-h^^^^x^+  .  .  . 
If 
when  n  is  any  negative  integral  exponent,  provided  that 

627.     W^hen   the    Exponent    is    Fractional.     I^et 

P 
n=~^  p  and  q  being  integers,  and  q  positive.     Consider 

the  series 


+  •  •  •       (1) 

This  series  is  seen  to  be  absolutely  convergent  when 
l>j;>— 1;  itisso  also  when  .ar=±l  and ->0,  but   it  is 


4S6  UNIVERSITY    ALGEBRA. 

somewliat  difficult  to  prove  this.  When  the  series  is 
absolutely  convergent  we  may  multiply  it  by  itself.  If 
we  square  it,  we  get 

q  \l 


f  (f-o-  ^  (f-O 


^^^        '         -^  -^'■+...  (2) 


f(?-)-(?-0 


+  -^^       ^    ^^^       -    "^-+---  (3) 

Observe  that  in  (2)  and  (3)  the  numerical  coefficient  of 
p  is  equal  to  the  exponent  of  Y.  Is  this  generally  true? 
If  it  is  true  when  the  coefficient  is  5,  then  we  can  show 
that  it  is  true  for  (^+1).     For,  multiply  the  members  of 

q  [2 

^q\q        )         \q 1^^  ^^^ 


by  the  members  of  (1),  and  get 

-f  1^  ^  +  .  .  . 

But,  since  the  coefficient  of  p  is  the  same  as  the  expo- 
nent of  y^  when  we  aibe  the  members  of  (1)  it  is  when  we 


BINOMIAT.  THEOREM.     ANY  EXPONENT.         457 

raise  them  to  the  fourth  power,  and  so  on.  Hence,  to 
raise  the  series  (1)  to  any  power,  a,  we  need  only  substi- 
tute ap  for  p. 

Raise  the  members  of  (1)  to  the  q  th  power,  then  the 
denominators  of  the  fractions  disappear,  and  we  have 

\r 

But  we  know  that  (l+^y=l+/-^+^^^5 — ^x'^^-^  .. 

Hence,  Y  is  one  of  the  ^th  roots  of  {X-^xy.  That  is, 
y=(l+:r)^.     It  follows,  therefore,  from  (1)  that 

-(--1) 
q  |Z 

provided  that  l>;t->— 1  or  that  Ji;=dbl  and  — >0. 

628.     To  expand  any  power  of  any   binomial,    say 
i^+yTy  we  write 

and  then  expand  (1  + -J  as  above. 

KXAMPIvKS. 

I.  (1-^)^  5.  (2+^r» 

2.  (1-3ji:)-3  6.  (l-:r2)-s 

3.  (l^Sxy  7.   (1  +  2^)"^ 

4.  (1—4;^;)"^  8.  (Aa-Sx)-^ 

9.  Show   that  only  two  terms  in  the  expansion   of 
(1—^)"^  have  a  positive  sign. 


45  8  UNIVERSITY   ALGEBRA. 

10.  The  coefficients  of  the  third  term  in  (l—x)^**  is  -J. 
Find  n  and  the  coefficient  of  the  fifth  term. 


II.  Find  the  rth  term  of 


(-5)-' 


12.  Find   the   coefficient   of  :r^    in   the  expansion  of 

13.  Ifx  is  very  small,  show  that  1—^x  is  an  approxi- 

mate  value  of ^- ' 

l  +  ;t:+l/l— :i; 


CHAPTER  XXVIII. 

CONTINUED  FRACTIONS. 

629.  Continued  fractions  have  already  been  mentioned 
in  Art.  179,  example  23,  but  in  this  chapter  we  desire 
to  study  this  kind  of  fractions  further  and  to  find  out 
some  of  their  principal  properties. 

630.  Starting  with  any  given  positive  number,  say  n, 
and  representing  the  integral  part  of  this  number  by  a^ 

we  evidently  have  n=a-\ — 

wherein  x^l,  • 

In  the  same  way,  representing  the  integral  part  of  x 

by  b,  we  have  x=b-\ — » 

y 

wherein  j^/>l. 

Similarly  we  have        y=c+—y 

u 


Replacing  x^y^  z-  -  -  by  their  values,  we  have 

'+7+., 

From  the  way  in  which  this  expression  i?  obtained  it 
is  evident  that  a  may  be  a  positive  whole  number  or  zero, 
but  b,  Cy  d'  '  '  must  be  positive  whole  numbers  at  least  as 
great  as  unity. 


460  UNIVERSITY   ALGEBRA. 

631.  Two  cases  of  continued  fractions  may  present 
themselves: 

First,  one  of  the  numbers  x^  y,  z,-  -  -  of  Art.  630,  may 
be  an  integer,  in  which  case  the  continued  fraction  termi- 
nates and  we  have  the  exact  expression  for  the  number  n 
in  the  form  of  a  continued  fraction. 

Second,  it  may  be  that,  however  far  the  above  operation 
is  carried,  none  of  the  numbers  jr,  j/,  ^,  •  .  .  are  integers. 
In  this  case  the  continued  fraction  will  never  terminate, 

but  if  one  of  the  fractions— '  —  >   — >    ...    of  Art.  630,  be 

X    y    z 

neglected  the  resulting  fraction  will  be  an  approximate 
value  of  the  number  n,  and  the  approximation  will  be 
closer  and  closer  as  the  neglected  fraction  is  farther  and 
farther  removed  from  the  beginning. 

632.  The  successive  approximations  found  as  just 
described  are  called  the  successive  Convergents  of  the 
continued  fraction. 

In  the  above  continued  fraction  a,  the  integral  part  of 

n,  found  by  neglecting  the  fraction-,  is  called  the  First 
\  X  1 

Convergent;  «  +  7>  found  by  neglecting  the  fraction -» 
o  1  ^ 

is  called  the  Second  Convergent ;  a-\ r,  found  by 

1  ^ 

neglecting  the  fraction  -,  is  called  the  Third  Conver- 
z 

gent,  and  so  on. 

633.  It  is  well  to  notice  that  when  a  number  is 
expressed  as  a  continued  fraction,  the  first  approximate 
value  of  that  number  is  not  necessarily  the  first  conver- 
gent. For  example,  if  we  express  .29  as  a  continued 
fraction,  we  readily  find 


CONTINUED   FRACTIONS.  461 

.29=^ 


2+i 


In  this  case  we  would  naturally  call  5  the  first  approx- 
imation to  the  number  .29,  but  in  this  case  the  first 
convergent  is  0;  z,  e.,  the  integral  part  of  .29,  but  the 

second  convergent  is  ^• 

It  will  now  be  readily  seen  that  in  case  of  any  positive 
number  less  than  unity,  the  first  convergent  is  0. 

634.  The  expression  of  any  negative  number  as  a  con- 
tinued fraction  will  be  readily  understood  from  a  special 

case:         -1.54=~2  +  .46=~24-— -j 

2  +  ^ 


5+^ 


Evidently  any  negative  number  may  be  treated  in  a 
manner  similar  to  that  in  which  —1.54  is  treated  in  this 
illustration.  Hence  the  expression  of  any  negative  number 
as  a  continued  fraction  may  be  made  to  depend  upon  the 
expression  of  some  positive  number  as  a  continued  frac- 
tion, and  it  is  therefore  not  necessary  to  consider  negative 
numbers  in  our  discussion  of  continued  fractions.  For 
this  reason  we  shall,  throughout  the  present  chapter, 
consider  all  the  numbers  with  which  we  deal  as  positive 
numbers  unless  the  contrary  is  expressly  stated.* 

635.  Theorem  I.  Every  commensurable  number 
corresponds  to  a  contmued  fraction  which  terminates  and 
conversely,  every  continued  fraction  which  terminates 
represe?its  some  commensurable  number. 


462  UNIVERSITY   ALGEBRA. 

m 
Let  —  represent  any  commensurable  number.     Divide 

m    hy   n   and   let   a   represent  the  quotient  and  r  the 

remainder.     Then  the  integral  part  of  —  is  a,  and  we  have 

m  r        .    \  ... 

n  n  fn\  ^  ^ 

Now  divide  n  by  r  and  let  b  represent  the  quotient  and 
Ti  the  remainder.     Then,  evidently, 

r  r  / ;  \  k^) 


(r,) 


Next,  divide  r  by  r^  and  let  c  represent  the  quotient 
and  ^2  the  remainder.     Then 

But  from  what  has  already  been  given,  it  is  evident 
that  the  successive  operations  in  this  process  are  pre- 
cisely those  in  the  process  of  finding  the  H.  C.  F.  of  m 
and  n,  and  if  these  operations  are  continued  long  enough 
there  will  come  a  time  when  some  division  is  exact  (even 
in  the  case  in  which  m  and  n  are  prime  to  each  other, 
in  which  case  the  last  divisor  is  unity)  and  the  process 
comes  to  an  end. 

If  now,  in  the  equation  (1)  we  substute  for  —  its  value 

T  ^ 

found  in  (2),  and  then  for  —  its  value  found  in  (3),  and 

so  on,  we  get  the  commensurable  number  —  expressed 

n 

as  a  continued  fraction  which  terminates. 

That  a  continued  fraction  which  terminates  represents 
a  commensurable  number  is  at  once  evident  when  the 
indicated  operations   are   performed,  for  in  this  case  a 


CONTINUED  FRACTIONS.  463 

fraction  is  finally  reached  whose  numerator  and  denomi- 
nator are  whole  numbers. 

To  fix  the  ideas  let  us  take  a  numerical  example. 
Suppose  we  have  the  continued  fraction 

1 


2  + 


4+i 


^4 


2+^=f.     Therefore  the  given  fraction  equals 


4+^==V'-     Therefore,  the  given  fraction  equals 

2+  1  -2+^-^^. 

636.  Theorem  II.  Every  incommensurable  number 
corresponds  to  an  unlimited  continued  fraction  and  con- 
versely, every  unlimited  continued  fraction  represents  some 
incommensurable  number. 

The  process  of  converting  a  number  into  a  continued 
fraction  (Art.  630)  applies  to  an  incommensurable  as  well 
as  to  a  commensurable  number,  but  if  an  incommensur- 
able number  be  converted  into  a  continued  fraction  the 
fraction  will  not  terminate,  for  if  it  did  terminate  it  would 
represent  a  commensurable  number,  by  theorem  I. 

Again,  as  every  commensurable  number  corresponds  to 
a  continued  fraction  which  terminates  ;  therefore,  a  con- 
tinued fraction  which  does  not  terminate  represents  a 
number  which  is  not  commensurable  ;  /.  ^.,  which  is 
incommensurable. 


4.64  UNIVERSITY   ALGEBRA. 

637.  Theorem  III.  In  every  continued  fraction  the 
values  of  the  convex  gents  are  alternately  less  and  greater 
than  the  value  of  the  continued  fraction  itself ;  the  first, 
third,  fifth,  etc.,  convergents  being  less,  and  the  second, 
fourth,  sixth,  etc,,  convergents  being  greater  than  the  con- 
tinued fraction. 

In  Art.  630  we  used  the  equations 

^=^+^  (^>1)  (1) 

^=^+i  (y>l)  ■                (2> 

y-'-^\  (^>1)  (3) 

z=d-v\  {u>V)  (4> 


u 


and  from  these  we  obtained  the  equation 

n=a^ 1^- 

bV--^  (5) 

Evidently  since  -  is  positive  a<^a-\ — ,  i.e.,  a<in,i.  e^ 

the  first  convergent  is  less  than  the  value  of  the  continued 
fraction. 

If,  for  X  in  (1),  we  substitute  its  value  obtained  from  (2) 

we  obtain  n=a-\ 

y 

Now,  since  3+-><^;  therefore --,-<t' 

y 

therefore,  a  -\ -7  <.a + -7- 

b^r- 

y 


CONTINUED    FRACTIONS.  465 

But  the  first  member  of  this  inequality  is  n,  which  is 
the  value  of  the  continued  fraction,  and  the  second  mem- 
ber is  the  second  convergent;  therefore,  the  second 
convergent  is  greater  than  the  value  of  the  continued 
fraction. 

Again,  if,  in  the  equation 

n=.a+ 


y 

we  substitute  for  _y  its  value  obtained  from  (3),  we  obtain 

2 

Now,  since  c-] — >^,  therefore  — t<— » 

c  +  - 
2 

therefore,  b-{ — — -  is  less  than  ^-| — ; 

c-\ — 
2 


therefore 


is  greater  than  :: 

b+- 
c 

therefore  a-\-- 


^+- 


•     2 


is  greater  than  a  H r 


c 


30  — U.  A. 


466  UNIVERSITY   ALGEBRA.^ 

But  the  first  of  these  last  two  expressions  is  7i\  i.e., 
the  value  of  the  continued  fraction  itself,  and  the  second 
is  the  third  convergent.  Therefore,  the  third  convergent 
is  less  than  the  value  of  the  continued  fraction  itself. 

Similar  reasoning  carried  to  any  convergent  desired 
shows  the  truth  of  the  theorem. 

Since  the  first,  third,  fifth,  etc.,  convergents  are  each 
less  than  the  value  of  the  continued  fraction  itself,  and 
the  second,  fourth,  sixth,  etc.,  convergents  are  each 
greater  than  the  value  of  the  continued  fraction  itself  it 
follows  that  the  contiyiued  fraction  is  intermediate  in  value 
between  any  two  successive  convergents. 

638.  Theorem  IV.  In  any  contimied  fractio7t  any 
convergent  is  intermediate  in  value  betweeji  the  two  immedi- 
ately preceding  convergents. 

Any  convergent,  say  the  rth  convergent  of  any  con- 
tinued fraction,  is  itself  a  continued  fraction,  all  of  whose 
convergents  up  to  the  ;th  are  exactly  the  same  as  the 
corresponding  convergents  of  the  given  continued  fraction, 
and  by  theorem  III  the  value  of  this  continued  fraction 
is  intermediate  between  any  two  of  its  successive  conver- 
gents. Therefore,  this  continued  fraction  is  intermediate 
between  the  (r— 2)d  and  the  (r— l)st  convergents,  and 
therefore  the  rth  convergent  of  any  continued  fraction  is 
intermediate  in  value  between  its  (r— 2)d  and  (r— l)st 
convergents  as  stated  in  the  theorem. 

Illustration.     The  continued  fraction 

■   1+^ 

3+-^  (1) 


4+ 


^-\ 


.M 


CONTINUED    FRACTIONS.  467 


has  for  its  fifth  convergent 


2+-^  (2) 

Now  it  is  evident  by  inspection  that  the  first,  second, 
third,  fourth,  and  fifth  convergents  of  (1)  are  respectively 
the  same  as  the  first,  second,  third,  fourth,  and  fifth  con- 
vergents of  (2)  (the  fifth  convergent  of  (2)  being  (2) 
itself). 

Since  by  theorem  III  (2)  is  intermediate  in  value 
between  its  third  and  fourth  convergents,  therefore  the 
fifth  convergent  of  (1)  is  intermediate  in  value  between 
its  third  and  fourth  convergents. 

By  calculating  the  values  of  the  convergents  of  (l),we 
readil}^  find 

10      291G0 
third  convergent    =--=^g^ 

f      ,u  ^43      29197 

fourth  convergent  =3^=  20370 

...  .       139      29190 

fifth  convergent  =-- =  ^^^ 

639.  Theorem  V.  Iti  any  continued  fraction  the  odd 
convergents  taken  171  order  form  an  increasi7ig  series  and 
the  even  convergents  taken  in  order  forj?t  a  decreasing  series. 

By  theorem  III  the  odd  convergents  are  less,  and  the 
even  convergents  greater,  than  the  continued  fraction 
itself.  Hence,  any  odd  convergent  is  less  than  any  even 
convergent. 

Now  let  us  consider  any  odd  convergent,  say  the 
(2r+l)st  convergent.  This  is  less  than  the  2/th  con- 
vergent (since  the  2rth  is  an  eveji  convergent),  and  being 


468  UNIVERSITY    ALGEBRA. 

by  theorem  IV  intermedate  between  the  2rth  and  the 
(2r— l)st  convergents  it  must  be  greater  than  the 
(2r— l)st  convergent,  and,  since  this  is  true  for  any 
value  of  r  it  follows  that  the  odd  convergents  taken  in 
order  form  an  increasing  series.  Similarly  the  2rth  con- 
vergent (being  an  eve7i  convergent)  is  greater  than  the 
(2r — l)st  convergent,  and,  being  intermediate  between 
the  (2r— l)st  and  (2r— 2)d  convergents,  is  necessarily  less 
than  the  (2r— 2)d  convergent,  and  this  being  true  for  any 
value  of  r  it  follows  that  the  even  convergents  taken  in 
order  form  a  decreasing  series. 

640.  Since  the  even  convergents  are  always  greater 
than  the  continued  fraction  itself  and  taken  in  order  form 
a  decreasing  series,  therefore  the  successive  even  convergents 
approach  nearer  and  nearer  to  the  value  of  the  continued 
fraction  itself. 

Also,  since  the  odd  convergents  are  always  less  than 
the  continued  fraction  itself,  and,  taken  in  order,  form  an 
increasing  series,  therefore  the  successive  odd  convergents 
approach  nearer  and  nearer  to  the  value  of  the  continued 
fraction  itself, 

641.  Law  of  Formation  of  the  Convergents.  In 
the  equation 

n=a-\ 7 


e+,  ^ 

a,  5y  c,  d,  etc.,  are  called  the  first,  second,  third,  fourth, 
etc..  Incomplete  Quotients. 

By  actual  calculation  the  first  four  convergents  are 
found  to  be  as  follows  : 


CONTINUED   FRACTIONS.  469 

Theyfr^/  convergziit  is 


a 
a  or  :r- 

The  second  convergent  is 

,  1       ab-V\ 

The  third  convergent  is 

,1  ,       c  abc-^c+a 

c 
The  fourth  convergejit  is 

A  ,1  ,     ^^+1 

^-{ 7-  =  a-\ =  a-\- 


,  ,  1  '  d  bcd^b-^d 

^+;~1         ^^7dV\ 

^__  abcd-\-cd-\-ad^-ab-V\ 
~  bcd-\'b-\'d        ^' 

In  the  third  convergent  we  notice  that  the  numerator 
is  obtained  by  multiplying  the  numerator  of  the  second 
convergent  by  the  third  incomplete  quotient  and  adding 
the  numerator  of  the  first  convergent.  The  denominator 
also  ot  the  third  convergent  is  obtained  in  a  similar 
manner,  viz:  by  multiplying  the  denominator  of  the 
second  convergent  by  the  third  incomplete  quotient  and 
adding  the  denominator  of  the  first  convergent. 

Also,  in  the  fourth  convergent,  we  notice  that  the 
numerator  and  denominator  can  be  obtained  by  the  same 
law,  and  we  are  led  to  think  that  this  same  law  is  prob- 
ably general. 

We  shall  now  show  by  induction  that  this  law  is 
general. 

n      n  o     n 
Let  -7- »  -7*^  J  --  be  three  consecutive  convergents,  the 

last  of  which  is  derived  from  the  others  by  the  law  under 


470  UNIVERSITY    ALGEBRA. 

consideration,  and  let  q^,  q^,  q^  be  the  three  correspond- 
ing incomplete  quotients ;  /.  e. ,  if  -^-  represents   the  kWi 

convergent,    then    q^    represents    the    /^th    incomplete 
quotient. 

Because  the  law  holds  for  the  third  of  the  above  con- 
vergents,  therefore      ^3=^3^2  +  ^^! 

^3^^3^^2+^l 

^3      ^3^2  +  ^1 
Now  an  inspection  of  the  convergents  of  any  continued 

fraction  will  make  it  plain  that  we  may  pass  from  --  to 

to  the  next   convergent,   say  — -   by  changing   ^3  into 

1  ^4 

^sH —  where  q^  stands  for  the  next  incomplete  quotient 

^4 
after  q^. 

Making  this  change,  we  obtain 

(^<4H F2+^l        ^  .         ^  . 

^4  ^  V  q^l       ^  (^3^2+^^l)g'4+^^2  _   ^4^3 +^^2 

^4~/  ,     1\,      ,   .,    "(^3^2+ ^1)^4 +^2"  ^4^3 +^2    ' 


(^3+-y2+v/, 


Hence  the  law  holds  for  the  next  convergent  after  -—• 

Hence  if  the  law  holds  for  the  formation  of  some  conver- 
gent it  also  holds  for  the  formation  of  the  next  convergent. 
But  the  law  does  hold  for  the  formation  of  the  fourth 
convergent,  therefore  it  holds  for  the  fifth  convergent, 
therefore  for  the  sixth  convergent,  and  so  on;  hence,  the 
law  is  general. 

642.  Theorem  VI.  The  difference  between  any  two 
successive  co7ivergents  is  equal  to  unity  divided  by  the  product 
of  the  denomi7tators  of  the  tivo  convergents. 

The  difference  between  the  first  and  second  convergents 

ab+1             ab-j-1  —  ab      1 
__ a  = ^ J 


CONTINUED    FRACTIONS.  47 1 

The  difference  between  the  second   and  third  conver- 
gent s  is 

adc+c-\-a      ab-\-l  __^  ab'^c-{-bc+ab — ab'^c—ab—bc — 1 
bc+l  'b~  b{bc-\-V) 


bibc^-Y) 

We  notice  in  this  difference  that  the  numerator  is  — 1, 
but  this  we  ought  to  expect  for  we  know  that  the  odd 
convergents  are  less  than  the  even  ones.  The  theorem 
supposes  that  in  taking  the  difference  between  two  suc- 
cessive convergents  the  less  is  subtracted  from  the  greater, 
/.  e. ,  the  odd  convergent  is  always  the  subtrahend  if  the 
numerator  of  the  difference  is  unity.  If  we  indicate  the 
difference  between  two  successive  convergents,  not  know- 
ing which  is  the  odd  convergent,  the  numerator  of  the 
difference  is  -f  1  or  —1  according  as  the  subtrahend  is 
an  odd  or  an  even  convergent. 

Now  suppose  the  theorem  true  for  some  two  successive 

convergents,  and  let  them  be  represented  b}^  -—-  and  -~ 

di  d^ 

respectively,  then     -^ -7-  = 


d-^      d^       d^d^i 
,'.  n^d^  —  n2,d-^  =  d[zl. 

Now  let  -7-    represent   the   next   convergent   after   --- 

d^  «2 

and  let  q  represent  the  last  incomplete  quotient  which 

forms  part  of  -7-'  then  by  Art.  641 


d^      qd<2,-\-d^ 

'  '  d^      d.^       d^      qd^-\-d^ 
qn^d^+n^d^ — qnc^dc2, — n^d^, 722^1  —  ^i-^d^, 

^2^1 — ^1^2        -F-^ 


472  UNIVERSITY   ALGEBRA. 

Therefore,  if  the  theorem  holds  for  the  difference 
between  some  two  successive  convergents,  it  holds  for 
the  difference  between  the  next  two  successive  conver- 
gents, therefore  the  theorem  holds  for  the  difference 
between  any  tw^o  successive  convergents. 

643.  Theorem  VII.  All  the  convergents  are  irre- 
diicible  fractio7is. 

Let  -—  and  — -  be    any   two   successive   convergents, 

then  by  the  preceding  article  7i^d^—n^d^^=±\, 

Now  if  it  were  possible  for  n  ^  and  d^  to  have  a  common 
factor  other  than  unity,  that  factor  would  be  a  factor  of 
the  whole  left  number.  But  the  left  member  being  equal 
to  ±1  can  have  no  factor  other  than  unity.  Therefore, 
^1  and  d^  have  no  common  factor,  therefore  the  fraction 

— -  is  irreducible. 

71 

But  -— ■  is  a7iy  convergent,  hence  any  convergent  is  an 

irreducible  fraction,  hence  all  convergents  are  irreducible 
fractions. 

The  first  convergent  is  an  integer  number  or  zero,  but 
we  may  consider  either  case  as  a  fraction  whose  denomi- 
nator is  unity,  and  hence  the  first  convergent  is  no 
exception  to  the  theorem. 

644.  Theorem  VIII.  Any  convergent  is  nearer  the 
value  of  the  continued  fraction  than  any  previous  conve7gent. 

In    Art.    630,    the  expressions  a-f-?  b-\ — >    etc.,    are 

X         y 

called  'Complete  Quotients.  Plainly,  in  any  conver- 
gent, if  the  last  incomplete  quotient  be  replaced  by  the 
corresponding  complete  quotient,  the  result  is  the  con- 
tinued fraction  itself. 


CONTINUED    FRACTIONS.  473 

^7  7Z  7Z 

Now,  let  -,-'  3->  -,—  be  any  three  successive  conver- 

gents,  and  let  q  be  the  last  incomplete  quotient  which 

n  1 

forms  part  of  -%—  and  q-\ —  the  corresponding  complete 
a.^  r 

no      q7i<y-\-n. 
quotient,  then  we  have  t-=     /  ,    .  • 

3         2      2    "^      1 

Replacing  ^  by  ^H —  and  representing  the  continued 
fraction  by  n,  we  have 

(^+7)^2 +  ^h 


Therefore        -/  —  72=  -/  — 


(^  +  -^2  +  ^1 


z.  ^.,-7^  --;2= 


"^^  d^q+-^d,^d^ 


AlsoM— -T-=.      -,,  , 


(1) 


474  UNIVERSITY   ALGEBRA. 

71 -^d  2  —  71 2d  I  i  1 


d2\(q+~)d2+d,~^     d,^q+-^d^+d^ 


n^  ±1 


'■  '•' ""-  Tr  ,  r/   ,  i\ ,  ,  ,  1  (2) 


d,\(g+-)d^  +  dC\ 

Now,  since  q  is  at  least  1  and  r  is  positive,  therefore 

lq-\ — )>1.      Also,  if  q'  represents  the  last  incomplete 

quotient  which  forms  part  of  ~  and  -^   represents   the 

convergent  immediately  preceding  -— ,  then  by  Art.  641, 

d2  =  q'  d^+d\  and  as  all  these  numbers  are  positive,  and 
at  least  as  great  as  1,  it  follows  that  ^2^^i-     Now,  as 

{q-\ — )>1  and  as  d2~>dy,  plainly,  on  both  these  accounts 

the  second  member  of  (2)  is  less  than  the  second  member 
of(l). 

Therefore,  n—  ^<~  —n, 
d<2.      «i 

Therefore,  the  convergent  —^  is  nearer  the  value  of  the 

continued  fraction  than  -7^  is.      But  -~  is   any  conver- 

gent,  therefore  any  convergent  is  nearer  the  continued 
fraction  than  any  preceding  convergent. 

In  Art.  640,  we  found  that  the  odd  convergents  are 
less  than  the  continued  fraction,  and,  taken  in  order, 
approach  the  value  of  the  continued  fraction.  Also,  that 
the  even  convergents  are  greater  than  the  continued 
fraction,  and,  taken  in  order,  approach  the  value  of  the 
continued  fraction.     Article   640,  taken   in   connection 


CONTINUED  FRACTIONS.  475 

with  this  article,  shows  that  all  the  convergents  taken  in 
order  are  alternately  less  and  greater  than  the  continued 
fraction,  but  continually  approach  the  value  of  the  con- 
tinued fraction. 

KXAMPI^KS. 

339 

1.  Convert  the  fraction  ^^7^-^  into  a  continued  fraction. 

Zoo 

236 

2.  Convert  the  fraction  7^7^  into  a  continued  fraction. 

ooU 

126  . 

3.  Convert  the  fraction  :j-^^  into  a  continued  fraction. 

lol 

4.  Convert  3.1416  into  a  continued  fraction. 

5.  Find  the  successive  convergents  of  the  continued 

30 
fraction  which  is  equal  to  jo* 

6.  Find  the  successive  convergents  of  the  continued 
fraction  which  is  equal  to  t^t^- 

7.  If  V>  -r'  -j~  b^  ^^y  three  successive  convergents 

^1    "2     ^3 
of  a  continued  fraction  show  that  (71 3  —n^dc^^  —  {d^—d^n  o . 

8.  Prove  that  the  numerators  of  any  two  successive 
convergents  of  a  continued  fraction  are  prime  to  each 
other.  Prove  also  that  the  denominators  are  prime  to 
each  other. 

9.  If-/'  ^-'  —-  be  the  first  three  convergents  of  a 

^1     ^2     d^  ,       n.      71^         1  1 

continued  fraction  show  that  -/-  —  -,-=-rT  "~  TT  ' 

10.     If--)  ~-y  -7-'  -;- be  the  first  four  convergents  of 
di    ^2     ^3     ^4 
a  continued  fraction  show  that 

n^  __  n^  _  _1 1_    ,    _1 

d^       d^       ^1^2       ^2^3       ^3^4 


476  UNIVERSITY   ALGEBRA. 

645.  We  have  seen  that  any  commensurable  number 
may  be  expressed  as  a  continued  fraction  which  termi- 
nates, and  that  an  incommensurable  number  may  be 
expressed  as  a  continued  fraction  which  does  not  termi- 
nate. 

We  desire  now  to  consider  those  continued  fractions 
which  result  from  a  particular  kind  of  incommensurable 
numbers,  viz :  quadratic  surds,  and,  as  the  subject  offers 
some  difficulty,  we  will  take  first  a  particular  quadratic 
surd,  express  it  as  a  continued  fraction  and  notice  the 
form  of  the  result,  after  which  we  shall  be  better  able  to 
understand  the  general  case. 

Let  us  then  express jL„  as  a  continued  fraction. 

Since  the  integral  part  of  this  expression  is  1,  we  have 


V6-1 


2 


_l-(/6-l)(v^6+l)  6-^5"/ 

=i+2_v|::3==i+_L_  (2) 


21/6-3 


5       _         5(2t/6  +  3) _10l/G  +  15^2K^6  +  3 

2i/6-3~  (2l/6-3)  {2\/&+S)  1^  3 

=2+(^^^-2)=2+2J^^=2  +  -4-        (3) 

2J/6-3 


CONTINUED   FRACTIONS.  477 

3(21/6  +  3)  61/6  +  9      21/6  +  3 


(5) 


2T/6-3     (2V/6-3)  (21/6  +  3)  15  5 

=  l  +  (^_l)=l  +  ?l^=l^^       (4) 

21/6-2 

5      ^         5(21/6  +  2)         ^101/6  +  10^1/6  +  1 
21/6-2     (2/6-2)  (21^6 +  2)""        20        "~      2 

Now,  the  last  expression  in  (5)  is  exactly  the  expression 
we  started  with.  Hence,  to  find  its  value  in  the  form  of  a 
continued  fraction  we  would,  of  course,  go  through  with 
the  work  over  again  and,  of  course,  finally  arrive  at  the 
same  expression  again.  It  is  therefore  unnecessary  to  go 
through  any  more  numerical  work,  but  from  the  results 
already  obtained,  we  can  write  as  many  terms  as  we  please 
of  the  continued  fraction  which  represents  the  quadratic 

surd  — ^ 

To  write  this  continued  fraction  we  substitute  for  the 
denominator  of  the  last  expression  in  (1)  its  equal  ob- 
tained from  (2),  and  then  for  the  denominator  of  the  last 
expression  in  (2)  its  equal  obtained  from  (3),  and  so  on. 
Thus  we  obtain 


^-h 


2+? 


1  +  -. 
In  this  continued  fraction  it  is  to  be  noticed  that  the 


478  UNIVERSITY   ALGEBRA. 

incomplete   quotients   repeat   themselves  over  and  over 
again,  always  in  the  same  order,  1,  1,  2,  1. 

646.     If  we  represent  the  quadratic  surd  — ^r—  by  x 

and  notice  that  the  last  expression  in  equation  (5)  of  the 
preceding  article  is  x  we  may  write 

1  +  — 1 

X 

Reducing  the  second  member  to  a  common  fraction,  we 

7;tr-f5 


have 

^    4x+3 

Hence, 

^x''-\-Zx=lx^h 

.'.  Ax'^—Ax=b. 

Solving  we  get 

l±l/6 

From  this  w^e  see  that  the  above  continued  fraction  is 
one  of  the  roots  of  a  certain  quadratic  equation. 

We  shall  see  presently  that  the  same  thing  is  true  of  a 
whole  class  of  continued  fractions. 

647.  When,  in  an  unlimited  continued  fraction,  a 
certain  number  of  the  incomplete  quotients  continually 
recur  in  the  same  order,  the  fraction  is  called  a  Periodic 
Continued  Fraction. 

The  incomplete  quotients  which  recur  constitute  the 
Period. 

When  the  period  begins  with  the  first  incomplete 
quotient  the  fraction  is  called  a  Simple  Periodic  Con- 
tinued Fraction,  but  when  one  or  more  incomplete 
quotients  occur  before  the  period  begins  the  fraction  is 
called  a  Mixed  Periodic  Continued  Fraction. 


CONTINUED   FRACTIONS.  479 

For  example : 

is  a  simple  periodic  continued  fraction  whose  period  is 
formed  from  the  incomplete  quotients  a,  b,  c. 

b+ 


is  a  mixed  periodic  continued  fraction  whose  period  is 
formed  from  the  incomplete  quotients  e.  f. ,  and  in  which 
the  incomplete  quotients  a,  b  occur  before  the  period 
begins. 

648.  Theorem  IX.  Every  simple  periodic  continued 
fraction  is  a  root  of  a  quadratic  equation  with  rational  coeffi- 
cients whose  roots  arc  of  opposite  signs. 

Let  -p^  and  --  be  the  last  two  convergents  of  the  first 
dy_^  d^ 

period  of  a  simple  periodic  continued  fraction,  and  let  x 

be  the  value  of  the  continued  fraction.  Then  by  Art.  641 

___xnr'\-nr^-^ 

from  which  we  readily  obtain 

drX'^  +  {dr-^—n^x—Ur-x^^. 
Now,  as  dr,  dr^^,  Ur,  n^-i  are  all  rational,  the  coeffi- 
cients of  this  equation  are  rational,  and  as  the  coefficient 
of  ;t:^  is  positive  and  the  absolute  term  negative,  the  roots 
of  this  equation  are  of  opposite  signs.     (See  Art.  290). 


^SO  UNIVERSITY    ALGEBRA. 

EXAMPLES. 

1.  Convert  — ^ —  into  a  continued  fraction. 

2.  Convert p —  into  a  continued  fraction. 

5 

3 

3.  Convert 7=  into  a  continued  fraction. 

4  +  1/3 

4.  Find  the  quadratic  equation  one  of  whose  roots  is 
the  value  of  the  simple  periodic  continued  fraction  whose 
period  has  the  incomplete  quotients  1,  2,  3,  4. 

5.  If—  and  ~  represent  respectively  the  third  and 
fourth  convergents  of  the  continued  fraction  in  example 


4,  express  —  and  ~  as  continued  fractions. 
7Z^  d^ 


6.  Find  the  quadratic  equation  one  of  whose  roots  is 
the  value  of  the  simple  periodic  continued  fraction  whose 
period  has  the  incomplete  quotients  4,  3,  2,  1. 

7.  Find  how  the  roots  of  the  equation  found  in  ex- 
ample 6  compare  with  those  of  the  equation  found  in 
sxample  4. 

8.  Find  the  quadratic  equation  one  of  whose  roots  is 
the  value  of  the  mixed  periodic  continued  fraction 

2- 


5+1- 


2-fl- 


^+-1 
^+3— 


CONTINUED    FRACTIONS.  48 1 

9.   Find  the  quadratic  equation  one  of  whose  roots  is 
the  value  of  the  mixed  continued  fraction 
1 


5  + 


3+- 


2+' 


3+. 

10.  Do  the  roots  of  the  equation  found  in  example  8 
have  the  same  or  opposite  signs  ? 

11.  Do  the  roots  of  the  equation  found  in  equation  9 
have  the  same  or  opposite  signs  ? 

12.  Convert  V  5  into  a  continued  fraction. 

13.  Convert  1^10  into  a  continued  fraction. 

14.  Convert  1^2  into  a  continued  fraction. 

15.  What  kind  of  continued  fractions  are  obtained  in 
examples  12,  13,  14? 

31  — U.  A. 


CHAPTER  XXIX. 


DERIVATIVES. 


649.  Notation.  A  definition  of  a  function  of  a  quan- 
tity was  given  in  Art.  358. 

To  represent  a  function  of  a  quantity  we  enclose  in  a 
parenthesis  the  letter  which  represents  the  quantity,  and 
write/  or  F  or  some  other  functional  symbol  before  the 
parenthesis,     e.  g, 

f{x),  F(^x),  F^{x)  denote  functions  oi x. 

/OO'  ^(j)y/i(y)  denote  functions  of y. 

f{x-\-Ji),  F{x+h'),f{x-\-]i)  denote  functions  of  ji:+/2. 

/(^),  Fid),  ^\f{a)  denote  functions  of  a. 

The  student  must  be  careful  not  to  look  upon  the  ex- 
pression fix)  as  meaning  /  times  x.  The  symbol  /,  as 
used  here,  is  not  a  multiplier  at  all,  but  simply  an 
abbreviation  for  the  words  function  of 

It  frequently  happens  that,  in  the  same  discussion,  we 
wish  to  refer  to  different  functions  of  x,  in  which  case  we 
use  different  functional  symbols,  as  F(^x),  f{x~),  f^{x), 
fix),  FXx),  <^(^),  lAW,  etc. 

It  also  frequently  happens  that,  in  the  same  discussion, 
we  wish  to  refer  to  the  same  function  of  different  quanti- 
ties, in  which  case  we  use  the  same  functional  symbol 
before  the  parenthesis  but  different  letters  within  the. 
parenthesis,  e.  g .  \if{x)  denotes  x'^  -\-\  then  /{a)  denotes 
«^-fl,  f{z)  denotes  ^^-fl,  etc.,  and  if  F^x)  denotes 
V x-\-Z,  then  F(^y)  denotes  Vjz-fB,  F{x-\-1i)  denotes 
V x-\-h'\-Z,  etc. 


DERIVATIVES.  483 

A  function  of  two  quantities  is  any  expression  in  which 
both  the  quantities  appear. 

To  represent  a  function  of  two  quantities,  say  ;r  andjj^, 
we  enclose  x  and  _y  in  a  parenthesis,  separated  by  a 
comma,  and  write  the  letter  /  or  /^  or  <^  or  some  other 
functional  symbol  before  the  parenthesis,  e.  g.  f{x,  j/), 
F{x,y),  ^{x,y),  etc. 

650.  In  such  an  expression  as  f{x,  y),  the  x  and  y 
are  entirely  unrestricted  in  value  and  independent  of  each 
other;  but  if  we  have  an  equation  like/(jf,  jr)=0,  then  x 
and  y  are  to  some  extent  restricted;  any  value  may 
indeed  be  given  to  one  of  the  quantities  but  then  the 
equation  fixes  the  value  of  the  other,  or  in  other  words, 
either  one  of  the  quantities  x  or  y  depends  upon  the  other 
one.  For  example:  if /(^,  jv)  stands  for  ;ir— j/4-2,  then 
when  this  is  not  put  equal  to  anything  there  is  no  rela- 
tion between  x  and  y.  We  may  let  x=^S  and  j/=5  or  7  or 
10  or  any  other  number.  But  if  we  put  this  same  function 
equal  to  zero,  f/ien  there  is  some  relation  between  x  and 
y  and  they  are  to  so77te  extent  restricted  in  value.  We 
may  let  ;t:=3,  but  then  j/=5  and  nothing  but  5. 

651.  If  the  equation  F{x,  >')=0  can  be  solved  for  y, 
we  can  express  y  in  terms  of  x,  or  y  can  be  determined 
as  a  function  of  x.     If  w^e  thus  determine  y  we  have 

In  this  equation,  y'=f{x),  we  may  look  upon  Jtr  as  a 
variable,  and  of  course  if  x  varies  y  will  also  vary.  We 
may  consider  x  to  vary  in  any  way  we  please,  but  then 
the  equation  determines  the  way  in  which  j/  varies.  For 
this  reason  ^is  called  the  Independent  Variable,  and  j^, 
which  is  the  function  of  x^  is  called  the  Dependent 
Variable. 


484  UNIVERSITY   ALGEBRA. 

652.  In  the  equation  of  jr=/(-^),  if  a  value  be  given 
to  X,  then  y  will  have  some  corresponding  value,  and  if 
X  be  given  another  value  different  from  the  first  one  then 
y  will  have  some  value  different  from  the  one  it  had  at 
first.  Moreover,  the  amount  by  which  y  thus  changes 
in  value  will  depend  in  some  way  upon  the  amount  by 
which  X  changes,  or,  in  other  words,  there  is  some  rela- 
tion connecting  the  change  in  the  value  of  y  with  the 
change  in  the  value  of  x.  This  relation  we  shall  examine, 
and  it  will  be  found  to  be  a  very  important  relation  in 
all  that  follows. 

653.  Suppose /(;i;)  to  stand  for  2x  +  4,  then  putting 
this  equal  to  y^  we  have 

y=2x+4:. 
Let  us  now  give  to  ;r  a  series  of  values,  say  the  success- 
ive integers  from  1  to  10,  and  in  each  case  compute  the 
corresponding  value  of  y.     The  results  may  be  expressed 

in  the  form 

jK  I  6     8     10     12     14     16     18     20    22     24 
^1  123456789      10 
where  any  number  in  the  lower  line  is  one  of  the  values 
of  X  and  the  number  immediately  above  it  is  the  corres- 
ponding value  of  y. 

If  x=  2  the  corresponding  value  of  j/  is  8, 
and  if  ;t:=10  the  corresponding  value  of  j^  is  24, 
and  if  :t:  be  considered  to  increase  from  2  to  10  then  at 
the  same  time  y  will  increase  from  8  to  24,  or,  starting 
Sitx=2,  \i  X  increases  by  8,  y  will  increase  by  16,  or  if 
the  increase  of  x  is  8,  the  corresponding  increase  of  _y 
is  16. 

Still   starting   at   x=2,  let  us  increase  x  by  various 
amounts  and  determine  the  corresponding  increase  oiy. 
The  results  may  be  arranged  in  the  form 

increase  of  jK  I  16     14     12     10     8     6     4     2 
increase  of  ;f  I    87       6       54321 


DERIVATIVES.  485 

If  we  had  started  with  some  other  value  of  x  than  2 
we  would  have  obtained  similar  results.  In  every  ob- 
served case  we  see  that  the  increase  oiy  is  just  twice  the 
increase  of  x,  or  in  every  observed  case 

increase  of  j/  _^ 

increase  of  x 
It  is  easy  to  see  that  this  is  necessarily  the  case  what- 
ever the  value  of  x  with  which  we  start  and  whatever 
the  amount  b}^  which  x  is  increased,  for  if  x  increases  by 
any  amount,  2x  will  increase  by  just  twice  that  amount 
and  the  change  in  the  value  of  x  does  not  affect  the  4, 
therefore  2;t:+4,  or  jk,  will  increase  twice  as  much  as  x 
increases,  or 

increase  of  j/  _  ^ 

increase  of  ;t:        ' 

654.  Notation.  In  what  follows  we  deal  largely 
with  equations  formed  by  putting  y  equal  to  a  function 
of  X,  and  as  we  shall  make  extensive  use  of  the  increase 
in  the  value  of  x  and  the  corresponding  increase  in  the 
value  of  y  it  is  well  to  have  a  convenient  notation  by 
which  these  amounts  of  increase  are  denoted.  So  in 
future  we  shall  use  A:r  to  denote  the  increase  in  the  value 
of  X  and  Aj  to  denote  the  corresponding  increase  in  the 
value  of  y. 

In  this  notation  the  fraction  at  the  end  of  Art.  653 
would  be  written 

The  student  is  cautioned  not  to  think  ot  A;t;  as  being 
A  times  x  for  the  symbol  A  as  here  used  does  not  stand  for 
the  coefficient  of  x  at  all,  but  simply  for  the  words  in- 
crease of. 


486  UNIVERSITY   ALGEBRA. 

655.     Let  us  now  consider  the  equation 
j/=^'-i-f  1. 

In  this  equation  give  x  the  successive  integer  values 
from  —3  to  7  and  compute  the  corresponding  values  of  j/. 
We  may  arrange  the  results  as  in  Art.  653. 

j^l     10      5      2     1     2     5     10     17     26     37     50 
^1-3-2-1012       3      4      5      6  ~~7 

If  x=^\  the  corresponding  value  of  jr  is  2,  and  if  x=7 
the  corresponding  value  of  y  is  50,  and  if  x  be  supposed 
to  increase  from  1  to  7,  at  the  same  time  y  will  increase 
from  2  to  50,  or  starting  at  x=\,  \i  x  increases  by  6  then 
y  will  increase  by  48,  or  when  A;t;=6,  Aj/=48. 

Still  starting  at  x—\,  let  us  give  t^x  various  values 
and  determine  the  corresponding  values  of  l^y. 

The  results  may  be  arranged  in  the  form 


Aj 

48 

35 

24 

15     8     3 

^x 

6 

5 

4 

3     2     1 

Ai/ 
Here  we  have  a  case  in  which  the  ratio  ~-  is  not  al- 

^x 

ways  the  same  as  was  the  case  in  Art.  653,  but  at  one 

time  it  is  -V-,  or  8,  at  another  time  it  is  ^f-y  or  7,  etc. 

Ai/ 
As  can  be  seen  by  the  above  scheme,  the  fraction  ~- 

^x 

takes  successively  the  values  8,  7,  6,  5,  4,  3  as  A;r  takes 
the  successive  values  6,  5,  4,  3,  2,  1. 

We  now  give  to  x  values  intermediate  between  1  and  2 
and  compute  the  corresponding  values  of  _>'.  The  results 
may  be  arranged  in  the  form 

y  I  2  2.0000200001    2.00020001    2.002001    2.0201    2.21 
x\  1  1.00001  1.0001  1.001  1.01        1.1 

As  before,  let  us  start  at  :r=l  and  give  to  t^x  various 
fractional  values  and  determine  the  corresponding  values 
of  Aj/. 


DERIVATIVES.  487 

The  results  may  be  arranged  in  the  form 

y  I  .21     .0201     .002001     .00020001     .0000200001 

x\  .1     .01      .001        .0001         ;ooooi 

An  examination  of  this  scheme  shows  that 

whenAjr=.l,  then  ~^=  ~  =2.1 
A;t-        .1 

when  A;r=.01,  then  ^=  :^?i  =2.01 

A;t-  .01 

when  A^=.001,  then  ^  =  •™-^  =2.001 
A^  .001 

1,       A  nnm    .t,       ^y       .00020001     ^^^^, 

when  A;i:=.0001,  then  -^  =  —  =2.0001 

t,       A         nmm    .-u       ^J      .0000200001     ^  ;,^^^, 

when  A^-=. 00001,  then  -f-  = ^^^^^^ —  =  2.00001 

A;r  .00001 

From  the  first  part  of  the  article  it  appears  that  —  is 
a  variable,  and  from  w^hat  we  have  just  obtained  it  fur- 
ther appears  that  as  ^x  is  taken  smaller  and  smaller  the 

Aj/ 
fraction  -^-  approaches  nearer  and  nearer  the  value  2,  or 

Ay 
in  other  words,  it  appears  that  the  fraction  ~-  approaches 

2,  i.  e.,  2  times  1,  as  ^x  approaches  zero. 

In  obtaining  the  result  it  is  to  be  noticed  that  we  con- 
sider X  to  increase  frotn  the  value  1,  but  if  we  let  x 
increase  by  various  amounts,  beginning  to  count  the 
increase  in  x  from  the  valued,  reasoning  exactly  as  we 

have  just  done  would  lead  to   the    conclusion  that  ~ 
^  A.V 

approaches  4,  i.  c,  2  times  2,  as  t^x  approaches  zero. 

Again,  if  we  begin  to  count  the  increase  in  x  front  the 

value  3,  reasoning  as  above  would  lead  to  the  conclusion 

Ai/ 
that  -^  approaches  6,  /.  ^.,  2  times  3,  as  A^  approaches 

zero. 


488  UNIVERSITY   ALGEBRA. 

656.  In  general,  if  a  be  taken  as  the  value  of  :r  from 
which  we  begin  to  count  the  increase  of  x  analogy  would 

lead  us  to  expect  that  the  fraction  -~  approaches  2<^   as 

A;r  approaches  zero,  or,  using  the  notation  of  Art.  432, 
limit  /Aj^/\ 

This  we  will  now  prove. 

Since  j/=x2  +  l,  (1) 

whatever  value  be  assigned  to  x  the  equation  will  enable 
us  to  compute  the  corresponding  value  of  y. 

First,  let  x—a  and  represent  the  corresponding  value 
of  J/ by  ^,  then  b^a'^  +  X,  (2) 

Now  let  x=a-\-^x 

then,  representing  the  corresponding  value  oiy  by  b-\-L^y, 
from  equation  (1)  we  get 

^+A^=(«  +  A^)2  +  1,  (3) 

or  simplifying, 

^  +  Aj^/=.^2^2aA;t:+(Ax)2  +  l.  .(4) 

Subtracting  (2)  from  (4),  we  get 

A_>/=2aA:r+(A;r)2.  (5; 

Dividing  (5)  by  A;t:,  we  obtain 

^=2^  +  A;r,  (6) 

As  ^x  varies,  of  course  the  two  sides  of  equation  (C) 
are  variables,  and,  indeed,  they  are  two  variables  that  are 
always  equal,  and  as  A:r  approaches  zero  these  two  vari- 
ables approach  limits. 

Hence,  by  chapter  XVIII,  Art.  420,  their  limits  must 
be  equal. 

limit  r.^)^2a. 


Hence,  A;^- OVA;^. 

Aj 
If  we  attempt  to  find  the  limit  of  — 

numerator   and   denominator   separately    we    are    led   to    ths    form 


A7 
If  we  attempt  to  find  the  limit  of -r—  by   finding   the   limit  of  the 


DERIVATIVES.  489 

0  Av 

7:  which  is  indeterminate.     We  must  therefore  take  the  fraction  -— 

0  A;c 

as  a  whole  and  not  try  to  find  the  limits  of  numerator  and  denomin- 
ator separately. 

Ay 

657.  The  value  of  the  fraction  -r-  when    that   frac- 

^^  Ay 

tion  is  constant,  or  the  limit  of  the  fraction  —-  as  Ax 

Ax 

approaches  zero  when  that  fraction  is  a  variable,  is  called 
the  Derivative  of  j' with  respect  to  x,  and  is  repre- 

dy 
sented  by  the  notation  — -  where  j^  is  a  function  of  x. 

dy 
Although  the  expression  ~r  is  written  in  the  form  of  a  fraction  the 

student  should  not  attempt  at  this  stage  to  assign  any  meaning  to 
numerator  and  denominator  separately,  but  should  look  upon  the 
expression  as  a  whole,  as  though  it  were  represented  by  a  single 
symbol.  When  the  subject  of  Differential  Calculus  is  reached,  the 
student  will  doubtless  find  a  meaning  for  dx  and  dy. 

658.  The  general  method  of  finding  the  derivative 
of  y  with  respect  to  x  is  that  used  in  Art.  656,  viz:  give 
to  X  some  value,  say  a,  arid  find  the  corresponding  value  of 
y;  then  give  to  x  a  new  value,  a  +  Ax,  and  again  find  the 
corresponding  value  of  y. 

Subtract  the  first  of  the  equations  thus  obtained  from  the 

secofid  and  the  result  will  be  the  value  of  Ay. 

A  y 
Divide  both  sides  by  Ax  and  the ;  esult  will  be  the  value  of  — ^  • 

•^  -^  Ax 

Finally,  fifid  the  limit  of  this  fraction  as  Ax  approaches 
zero. 

659.  We  will  now  illustrate  the  method  by  a  few 
examples. 

First.     Find^  whenj/=4.r2-f5.  (1) 

Let  x=a  and  represent  the  corresponding  value  of  y 
by  ^,  hence  ^=4^2+5.  (2) 


490  UNIVERSITY    ALGEBRA. 

Now  let  :r=a  +  A;r  and  the  corresponding  value  of  j/ 
will  be  the  value  b  plus  the  amount  by  which  y  has 
been  increased,  or  b  +  t^y,  hence 

^+Ajj/=4(a  +  Ajr)2  +  5.  (3) 

Expanding,     b-\-l^y=\a'^ +  %ab.x-\-^{l^xy -^h,  (4) 
Subtracting  (2)  from  (4),  we  get 

A_y=  8aAjr  +  4(A;r)  ^ .  (5) 

Dividing  (5)  by  ^x,  we  obtain 

^=8«  +  4(A^).  (6) 

Taking  the  limit  of  each  side  as  A,r  approaches  zero  we  get 

|-8».  (8) 

dy 
Second.     Find-f^- when  ^=^^2+^.  (1) 

Let  ^=<2  and  represent  the  corresponding  value  oi  y 
by  b,  then  b=ca''-' -{-e.  (2) 

Now,  letting  x=a-^^x  we  get 

b  +  Aj/=  ^(a  +  A;t:)  2  +  ^,  (3) 

or,  expanding, 

^+ Ajv=^a2  +  2«<rA;r+<A;t:)"  +^.  (4) 

Subtracting  (2)  from  (4),  we  get 

AjK=2^rAjt:+<Ajr)2.  (5) 

Dividing  both  sides  of  (5)  by  ^x,  we  get 

^-^lac^cl^x,  (6) 

Taking  the  limit  of  each  side  as  ^x  approaches  zero,  we 

limit  /Aa/\     ^ 
^^^  ^x-  o(a^)=2«^-  (7) 

g  =  2..  (8) 


DERIVATIVES.  491 

dy 
Third.     Find  ~  when  y^cx'^-^-ex  +f,  ( 1 ) 

Let  ;t:=«  and  represent  the  corresponding  value  of_y 
by  ^,  then  b=ca'^ -\-ea-\-f.  (2) 

Now,  letting  ;»;=«  + Ajt:,  we  get 

b+I^y=c{^a-\-^xy+e{^a  +  ^x)+f,  (3) 

or,  expanding  and  arranging, 

b-\-  ^y^ca"-  +ea+/^(2ac+e)^x-^ci^xy .  (4) 

Subtracting  (2)  from  (4),  we  get 

Ay=  (2ac-\-e)^x+c(i^x)  2 .  (5) 

Dividing  both  sides  by  ^x,  we  get 

-^  =  2ac^e-^c^x.  (6) 

Taking  the  limit  of  each  side  as  A;r  approaches  zero,  we 

nit  j 

::  0' 

dy 


limit /^J^^\     On.A..  n\ 

get  ^x-(kKxr^'''+'^  (^) 


^^=2..+  ..  (8) 

660.  In  what  precedes,  ^x  has  always  been  considered 
positive,  but  ^x  may  be  negative,  in  which  case  x  is 
increased  by  a  negative  quantity,  or  is  really  diminished, 
so  it  may  be  more  proper  to  call  Ajt:  the  chaiige  in  the 
value  of  X  than  to  call  it  the  amount  by  which  ;i;  has  been 
hicreased.  Either  way  of  speaking  is  proper,  provided 
we  understand  that  the  increase  may  be  negative.  In 
any  case  ^x  is  the  amount  that  must  be  added  to  one 
value  of  X  to  give  another  value  of  x,  and  if  the  second 
value  is  greater  than  the  first,  the  amount  added  will  be 
negative. 

The  same  remark  applies  to  Ajj/. 


492  UNIVERSITY   ALGEBRA. 

EXAMPI^ES. 

By  the  method  already  explained  and  illustrated  find 
the  derivative  of  the  following  expressions,  supposing  in 
each  case  that  a  is  the  value  of  x  from  which  the  increase 
of  X  is  counted  : 


I.     3a'+2. 

'  5-     cx'^ 

2.     Zx'^-\-2x. 

6.     c(x^\y. 

3.     {x+l){x+2). 

1 

7-     ^- 

4.    (.x+cy. 

8.     Vx. 

661.     Extension  of 

Meaning  of  -,-•    In  Art.  659, 

we  found  that  when  j'=4. 

x^+o, 

dx 

when                           y=i 

Tx'^+e, 

dx 

and  when            y=cx'  -f 

■ex+f, 

dx 

dv 
In  each  case  -f-is  of  coure  a  co?tstant  as  it  should  be  by 

^^  dy 

the  definition  in  Art.  657,  where  -f-  is  defined  by  a  limit, 

ax 

and  the  limit  of  a  variable  is,  by  definition,  a  constaiit. 

d  V 
In  each  case  here  noticed  -f-  is  a  constant  whose  value 

dx 

depends  upon  the  value  a  from  which  we  begin  to  count 

dv 
the  increase  of  x,  or,  as  we  may  say  -,-  is  a  function  of  a, 

dv  .  . 

while  in  Art.  654,  -;-  is  a  constant  which  does  not  de- 
dx 

pend  upon  a, 

*      dv 
In  any  case  — -  is  either  a  function  of  a  or  it  is  inde- 
pendent of  a,  and  when  it  is  a  function  of  a  the  a  is  the 
value  from  which  we  begin  to  count  the  increase  of  x. 


DERIVATIVES.  493 

Now,  as  we  may  begin  to  count  the  increase  from  a7iy 

value  of  X,  a  is  of  course  ajiy  value  of  x,  and  so  we  may 

d  V 
represent  it  by  x  instead  of  a  and  relieve  -r-  from  being  a 

d  V 
constant,  or,  in  other  words,  wherever  -/-  was  a  function 

dx 

of  a  by  the  original  definition  we  regard  it  now  and 
hereafter  as  the  same  function  of  x  that,  by  the  original 
definition,  it  was  of  a. 

662.  Let  us  now  work  out  a  case  that  was  worked  in 
Art.  659,  using  x  now  where  a  was  used  before. 

Take  y=Ax''  +  b.  (1) 

If  we  write  x-\-/:^x  in  place  of  x  and  therefore  y  +  ^y 
in  place  of  y  we  have 

jI/  +  A)/=4(;i;+A;t:)2  4-5  (2) 

Expanding, 

j^  +  A>/=4jt:2+8;rAx+4(A;t')2  +  5.  (3) 

Subtracting  (1)  from  (3),  we  obtain 

A_y=8;t:A;r+4(A;i;)2  (4) 

Dividing  both  sides  of  (4)  by  ^x,  we  get 

^=8;t-+4A;r.  (5) 

A;r  ^  ^ 

Taking  the  limit  of  each  side  as  ^x  approaches  zero,  we 

g=8..  (7) 

We  notice  that  the  result  is  exactly  the  same  as  equa- 
tion (8)  in  the  first  example,  under  Art.  659,  except  that 
X  appears  here  where  a  appeared  before. 

We   shall   hereafter  proceed   as  we  have  just  done. 

Usually-;^  will   be  a  function  of  Xy  but  occasionally, 

^^        dv 
as  in  Art.  654,  -^  will  turn  out  to  be  a  constant. 
dx 


494  UNIVERSITY   ALGEBRA. 

663.  Derivative  of  a  Constant. 

I^et  j=  a  constant.  Then,  as  x  does  not  appear  in  the 
expression  for  y,  any  change  in  the  value  of  x  does  not 
affect  y,  or  ^x  may  have  any  value  whatever,  but  Aj/  is 
always  zero. 

Hence  T^=0. 

t^x 

dy      ^ 
therefore  '3~=0. 

ax 

664.  Derivative  of  the  Sum  of  two  Functions 
of  X, 

Let  one  function  of  x  be  represented  by  ti  and  the  other 

by  V,  and  let  their  sum  be  represented  by  y\  then 

y=7i-\-v  (1) 

When  X  is  increased  by  Ax  suppose  that  7c  is  increased 

by  A?^,  V  is  increased  by  Az;,  and  j^  is  increased  by  Aj^/, 

then  after  x  is  increased  by  Aji^  we  have 

y-{-Ay—7c-\-A7C-\-v-i-Av,  (2) 

Subtracting  (1)  from  (2),  we  get 

Ay=A7i  +  Av.  (3) 

Dividing  both  sides  of  (3)  by  A;t:,  we  get 

^=^  +  ^.  a^ 

A;t;      A:r      Ajt:  ^  ^ 

Taking  the  limit  of  each  side  of  (4)  as  Ax  approaches 
zero,  we  get 

limit  (^\^  limit  (^\_i.  limit  (^_^\  .rx 

Ax  ::  0\AxJ     Ax:^0  \AxJ  "^  Ax  ^  0\AxJ  ^^ 

dy  __du     dv 
dx     dx     dx 
In  the  same  way  if  y=u—v  we  would  get 
dy     du     dv 
dx     dx     dx 
The  result  may  be  expressed  thus: 
The  derivative  of  the  algebraic  sum  of  two  f7cnctio7is  of  x 
equals  the  sum  of  their  separate  derivatives. 


DERIVATIVES.  495 

665.  Derivative  of  the  Sum  of  any  Number  of 
Functions  of  x, 

lyCt  there  be  any  number  of  functions  of  .r  represented 
by  u,  V,  w,  etc.,  and  let  their  sum  be  represented  by  jf; 
then  y=u-\-v-\-w-\-  •  •  •  (1) 

Increase  x  by  the  amount  A.r  and  suppose  that  u,  v, 
w,  etc.,  are  increased  by  the  amounts  t^tt,  Az;,  Aze',  etc., 
respectively  and  y  is  increased  by  Aj/,  then,  after  x  is 
thus  increased, 

j/+^y=2i-{-A2i-j-v-{-Av-\-w-\-Aw-{-  •  .  .  (2) 

Subtracting  (1)  from  (2),  we  get 

Aj/=A?^  +  A'z^H-Aze;-f  ...  (3) 

Dividing  both  sides  of  (3)  by  ^x,  we  have 
Ay^A^     A^      A^ 

/S.x     Ax^Ax  ^  Ax^  '  '  ^  ^ 

Taking  the  limit  of  both  sides  of  (4)  as  Ax  approaches 
zero,  we  have 

dy  _du     dv     dzv 
dx     ~dx     dx     dx 
If  some  of  the  signs  in  (1)  had  been  minus,  the  same 
process  could  have  been  applied  and  the  result  would  have 
had  minus  signs  in  the  same  positions  as  they  appeared 
in  the  original  functions. 

The  result  may  be  expressed  thus: 
The  derivative  of  the  algebraic  sum  of  several  finictions 
of  X  eqjials  the  algebraic  sum  of  their  separate  derivatives. 

EXAMPLES. 

Find  the  derivative  with  respect  to  x  of  the  following 
expressions: 

1.  Ix^-Y^x^^+x,  4.     x'-'  +  Zx—^ 

2.  X^+x'^+X+1,  5.       X'^—X-{~1. 

3.  x^  —  l.  6.     x^  +  1. 


496  UNIVERSITY    ALGEBRA. 

666.  Derivative  of  the  Product  of  two  Functions 
of  X. 

Let  2i  and  v  be  the  two  functions  of  x^  and  let  y  be 
their  product,  then 

y=2iv.  (1) 

Now  increase  x  by  ^x  and  suppose  the  corresponding 
amounts  by  which  y,  u,  and  v  respectively  increase  are 
Ay,  Az^,  and  Az;,  then 

j|/4-Aj/=(z/-fAzO(^^+Az;).  (2) 

Expanding  (2)  we  get 

y-\-^y=2iv  +  7i^^v-\-v^u  +  ^.u^.v,  (3) 

or  j/  +  A_y=2^^'  +  2/Az;+(z;+Az;)Az^.       ^  (4) 

Subtracting  (1)  from  (4),  we  get 

Aj/=z/A2;+(z^+Az;)Az/.  (5) 

Dividing  both  sides  of  (5)  by  A;tr,  we  get 

Ajt:         A;i;      ^  ^  ^x  ^  ^ 

Taking  the  limit  of  each  side  of  (6),  remembering  that 
the  last  term  of  the  right  hand  member  is  the  product  of 
two  variables,  hence  its  limit  equals  the  product  of  their 
separate  limits,  and  that  Az;  approaches  zero  as  A.r  ay^- 
proaches  zero,  hence  the  limit  oi  v-\-l^v=v,  we  get 

limit  fM=^  limit  ^  +  z.  limit  ^.  (7) 

Aj^::  ^\^xl  i^x  A^  "•  ^ 

dy  ^     (^,^du^  -.o>. 

dx         dx         dx 

667.  Derivative  of  the  Product  of  any  Number 
of  Functions  of  x. 

First,  take  three  functions  of  x,  say  u,  v,  and  Wy  and 
let  y  be  their  product,  then  we  have 

y=uvzv,  (1) 

Let  v'w^=v\  then  y=zw\  and  hence 

dy  __  ,du        dv'  ^^^ 

dx        dx        dx 


DERIVATIVES.  497 

^  dv'        dv   ,     dw  ,^. 

But  —-^w-^-\-v--'  (3) 

dx        dx       dx 

hence  by  substitution  in  (2)  we  have 

dy         dii      '     dv  ,       dw  ... 

U'.\'  U/.X'  Vt/^V'  U/.A' 

Now,  if  we  have  any  number  of  functions  of  x,  say  u, 

Vy  w,'  ■  '  and  if  we  let  y  be  their  product,  we  have 

y=-uvwz    •  •  (1) 

I^et  the  product  of  all  the  functions  after  the  first  be 

represented  by  a  single  letter,  that  is,  let 

v'=vwz-  .  . 

then  y=uv' . 

d  V 
Find  -J-  as  the  product  of  two  functions.     Then 

-T-=v'- — Vu-z-'  (2) 

J  I  dx        dx        dx 

Find  -7—  by  letting  v'=v2v\  where  w'  represents  the 

product  wz  -  ■  ' 

Substitute  the  value  thus  found  in  (2).     The  result 

will  contain  one  term  involvino^  -—  • 

^  dx 

Find  the  derivative  by  considering  w'  as  the  product 
of  two  factors. 

Continue  this  process  until  finally  the  product  of  the  last 
two  factors  of  the  expression  with  which  we  started  is^ 
reached. 

The  result  may  be  stated  thus : 

The  derivative  with  respect  to  x  of  the  product  of  miy 
number  of  functimis  is  equal  to  the  sum  of  all  the  products 
obtained  by  multiplying  the  derivative  of  each  factor  by  the 
product  of  all  the  other  factors . 

If  the  equation  here  described  be  divided  through  by 
the  product  of  all  the  given  functions,  the  result  may  be. 
represented  in  quite  a  convenient  form,  viz : 
1    dy ^\    du     1    dv      1    dw     1    dz 
y   dx     ti   dx     V    dx     w  dx      z   dx  '     '  ' 


49?  UNIVERSITY   ALGEBRA. 

KXAMPLKS 

Find  the  derivative  with  respect  to  x  of  the  following 
expressions  without  performing  the  multiplications  in- 
dicated : 

1.  (;t:— l)(:r— 2)  compare  with  example  4,  Art.  665. 

2.  {x'^—x+V){x-^V)  compare  with  example  6,  Art.  665. 

3.  (jr^+;r+l)(;r— 1)  compare  with  example  3,  Art.  665. 

4.  {x'^  +  \Xx^-\-ax+b'). 

668.  Derivative  of  the  Quotient  of  two  Func- 
tions of  X. 

Let  u  and  v  be  the  given  functions  of  x^  and  let  y  be 
their  quotient,  then 

y--  (1) 

From  (1),  by  multiplying  by  v,  we  get 

zc=vy,  (2) 

du       dy       dv  .^. 

^^''''^  'di'^'^dx^^Tx  ^^^ 

U/.\/  fAfJ\^  lA/.A' 

du  _  dy     u    dv 

dx       dx     V    dx  " 

Multiplying  both  sides  by  v,  we  get 

du^    ^dy       dv_^  , 

dx  dx       dx 

Transposing  and  dividing  by  z'^,  we  get 

du        dv 
dy      dx        dx 


du        dv  .^. 


dx  v'^ 

Expressed  in  words  this  is 

The  derivative  of  a  fraction  equals  the  deno7ninator  into 
the  derivative  of  the  mcmerator  minus  the  numerator  into 
tfie  derivative  of  the  denominator  all  divided  by  the  square 
of  the  denominator. 


DERIVATIVES.  499 

KXAMPI^KS. 

Find  the  derivative  with  respect  to  x  of  the  following 
expressions  : 

1. compare  wath  example  5,  Art.  665. 

X^ —^X'^ -\-\\x—^  .  .  1     ^     A    i.    nnr^ 

2.   _ compare  with  example  4,  Art.  665. 

x''-\ 
3. 


4. 


x  +  1 
x'^  +  l 


669.  Derivative  of  a  Function  of  Another  Func- 
tion of  X. 

Suppose  J/  is  some  function  of  z,  and  ^  is  some  function 
of  X,  then  ultimately  j/  is  a  function  of  Xy  hence  it  has  a 
derivative  with  respect  to  x. 

But  as  y  is  directly  a  function  of  ^  it  has  a  derivative 
with  respect  to  ^. 

Moreover,  as  -sr  is  a  function  of  x  it  has  a  derivative 
with  respect  to  x. 

We  have  identically 

Ax       Az '  Ax 
Taking  the  limit  of  each  side  as  Ax  approaches  zero, 
remembering  that  the  limit  of  the  product  of  two  vari- 
ables equals  the  product  of  their  limits,  we  have 

Hmit  (^\^  limit  {^\       limit  /^\  (2) 

Ax^  0\AxJ     Ax  :;  OVA-a'/  *  Ax  ^  0\AxJ 
Now  z  being  a  function  of  x,  we  may  write 

and  if  x  be  increased  by  Ax,  we  have 

2-\-Az=/(x-\-Ax'), 


5o6  UNIVERSITY   ALGEBRA. 

and  from  this  it  is  evident  that,  as  t^x  approaches  zero, 
A^  must  also  approach  zero.     Hence 

limit  /^^\       limit  l^\  (3) 

A^  -  OVaW     ^x  ::  OVA^y 
Substituting  from  (3)  in  (2),  we  have 

limit  {^^  limit  /^\       limit  l^_l\  (4) 

A;r  ::  OVAaV      A^  :;  0  V A^  y  *  A;t:  -  OV'A;^/ 

d  V 
The  left  member  of  (4)  is  -y-^  the   first   factor   of   the 

dy  ^^ 

right  member  is  -~^  for  it  is  just  the  same  as  the  left 

member  except  that  z  everywhere  takes  the  place  of  x  ; 

dz 
and  the  second  factor  of  the  righi  member  is  -^ — 

Hence  ?=??  (5) 

dx       dz  dx 

li  j/=z2^  and  z=x'^-i-2,  then 

dz  dx 

Hence  by  equation  (5) 

^^=2^.  2x=4zx=4x(x'^  +2)=:4:x^  +8x. 

It  is  easy  to  see  that  this  result  is  correct,  for  in  the 
equation  jy=z^,  substituting  the  value  of  z  we  have 

y:=(x'^  +2y=X^  +4:X^  +4:, 
^-=4:X^^SX. 

dx 
KXAMPLKS. 

Find  the  derivative  with  respect  to  x  of  the  following 
expressions: 

1.  (x'^+ax+dy. 

2.  (;r2  +  l)2  4-3(;i;2  4-l). 

3.  (x+ay  +  2(x+a-). 


DERIVATIVES.  5OI 

4.     (2x+Sy+5(2x+S)+4:. 

6.     2(jtr2-l)3  4-4(;i:2-l)2  +  (;r2-l). 

670.    Derivative  of  any  Positive  Integral  Power 

of  X. 

Let  y=x\  (1) 

Give  to  X  the  value  x-h^x,  then 

j/  +  Ay=(jt:+A;t:)".  (2) 

Expanding  the  right-hand  member  of  (2),  we  get 

Subtracting  (1)  from  (3),  we  get 
^y=.nx'--^Ax+''^'\~^K''-'-(r^xy-i-  ■  •  .  +(Axy\  (4) 

Dividing  both  sides  of  (4)  by  A;r,  we  get 

^=nx''-^  -f '^fc^^^"-'Ax+  .  .  .  +(A;r)"-i.  (5) 

Taking  the  limit  of  each  side  as  Ax  approaches  zero,  we 

have  ^=nx"-\  (6) 

Reasoning  exactly  as  above  we  could  show  that  when 
y=ax'\ 

ax 
This  formula  may  be  expressed  in  w^ords  thus: 
T/ie  derivative  with  respect  to  x  of  ax"  is  equal  to  the 
t>roduct  of  the  expojient,  the  coefficient,  and  x  with  the  expo- 
nent diminished  by  one^ 

It  is  to  be  noticed  that  this  formula  applies  to  the 
derivative  with  respect  to  jr  of  a  power  of  x.  Of  course 
any  other  letter  could  be  used  as  well  as  x  to  denote  a 
variable. 

Thus,  when  r=«y,  -^-^naz**''^ . 
"^  dz 


502  UNIVERSITY   ALGEBRA. 

But  we  must  be  careful  not  to  use  this  formula  to  find 
the  derivative  with  respect  to  some  quantity,  of  a  power 
of  some  other  quantity,  or,  in  other  words,  in  order  to  be 
able  to  use  this  formula  the  quantity  which  is  raised  to  a 
power  must  be  the  same  as  that  with  respect  to  which  the 
derivative  is  taken. 

671.     Derivative  of  any  Negative  Integral  Power 
of  X, 
I.et  _;,=^-=l.  (1) 

g=^^;^  by  Art.  668.  (2) 

Simplifying,  remembering  that  the  derivative  of  a  con- 
stant is  zero,  we  get 

dy.         nx''~^  „  , 

dx  x^'' 

It  may  be  objected  to  this  method  that  we  have  used 
the  formula  for  the  derivative  of  a  fraction  w^hose  numer- 
ator is  1  when  that  formula  supposed  that  numerator  and 
denominator  were  each  functions  of  x. 

X 
We  may,  then,  take     y——;^^x 

and  now  using  the  formula  of  Art.  ^^'^,  we  get,  as  before 

dx 
It  easily  follows  that  if 

y:=ax~**. 

then  -~  =  —nax  ""  ^. 

dx 

Here,  as  in  Art.  G70,  in  order  to  use  the  formula,  the 

quantity  raised  to  a  power  must  be  the  same  as  the  one 

with  respect  to  which  the  derivative  is  taken. 


DERIVATIVES.  503 

We  may  express  this  formula  in  words  thus: 
The  derivative  with  respect  to  x  of  ax"''  is  equal  to  the 
prodzict  of  the  exponent,  the  coefficient,  a7td  x  with  its  expo- 
nent diminished  by  one, 

672.     Derivative  of  a  Fractional  Power  of  x. 

p 


Let    . 

z=x"^ 

(1) 

then  let 

y^Z^^X^, 

(2) 

Then 

J'=$'.J-.  Art.669. 

dx     dz   dx                                 • 

(3) 

But 

g=^-. 

(4) 

hence 

dy        ^  .     dz 
dx     ^         '    dx 

(5) 

But  from  (2) 

dx     ^^        ' 

(6) 

hence  from  (5) 

and  (6)    -   # 

(7) 

Dividing  by  qz'^ 

',  which  is  the  same  gx^,  we  get 

,dz      p     , 
dx      a 

(8) 

Multiplying  the  left  member  by  z  and  the  right  member  by 

p 
x'q,  which  is  the  equal  of  z,  we  obtain 

^dz     p  l-i 

— -  =  -X9 

dx      q 

t 
The  same  reasoning  would  show  that  \i  y=^axq,  then 

dy     ap  ^ -■[ 

dx       q 

Hence,  as  in  the  two  preceding  articles, 

The  derivative  with  respect  to  x  of  axg  is  eqiial  to  the 
product  of  the  exponent,  the  coefficient,  and  x  with  its  expo- 
nent diminished  by  07ie, 


504  UNIVERSITY    ALGEBRA. 

BXAMPIvKS. 

Find  the  derivative  with  respect  to  x  of  the  following 
expressions: 

I.      (a-2-^+l)-f2(jr2-:t:+l). 

(x^-\\  ,  Jx^-\\ 


-  i^H"^) 


Jx'^^1 

i>V-i 


5.    (x+vT^^Y- 
Vi+x-^vT^x 

vY+x—Vr^x 

(     ^__  y 


8.      {a^+J)^a^  +  x^^ 

Va-i^x 


9. 


10. 


Va+  }/x 


11.  -xla-i h--^- 

\         X     x^ 

12.  {a-^xy.(d-\-xy. 


CHAPTER  XXX. 

INCOMMKNSURABI.K  KXPONKNTS  AND  LOGARITHMS. 

673.  We  have  found  before  that  whatever  commen- 
surable numbers  be  represented  by  7i  and  r,  the  following 
laws  of  exponents  hold : 

(^«)-=^'-  \b)  ~~y 

but   no   meaning  has  yet  been  given  to  numbers  with 
incommensurable  indices. 

A  number  raised  to  any  power,  as  a  above,  is  called  a 
Base.  The  present  discussion  is  confined  to  the  case  in 
which  the  base  is  positive. 

674.  A  given  base,  affected  by  an  exponent,  may 
have  more  than  one  value,  as  for  instance:  (25)2"=db5, 
but  a  base  affected  by  any  commensurable  index  has  among 
its  values  one  which  is  positive. 

For  any  integral  power  of  a  positive  base  is  evidently 
positive.  Any  root  of  a  positive  base  has,  among  its 
values,  one  which  is  positive,  and  since  any  power  of  this 
i;oot  is  positive,  therefore  a  positive  base  affected  by  a 
positive  fractional  index  has  among  its  values  one  w^hich 
is  positive.  Again,  since  a  positive  base  affected  by  a 
negative  index  is  the  reciprocal  of  the  same  base  affected 
by  a  positive  index,  therefore  a  positive  base  affected  by 
a  negative  fractional  index  has  among  its  values  one 
which  is  positive.  This  positive  value  is  all  that  is  con- 
sidered in  the  present  discussion.     So  that  whenever  we 


506  UNIVERSITY   ALGEBRA. 

deal  with  an  expression  like  a''  in  the  present  chapter 
both  a  and  a''  are  positive.  These  restrictions  must  not 
be  lost  sight  of. 

675.  What  meaning  must  a  base  affected  j with  an 
incommensurable  exponent  have  and  be  consistent  with 
the  meanings  already  assigned  to  commensurable  expo- 
nents ?  For  example  :  what  is  the  meaning  of  10^  ^  ? 
The  exponent  V  2  does  not  show  how  many  times  10  is 
used  as  a  factor,  for  t/2  is  not  a  whole  number,  and  it 
does  not  indicate  a  power  of  a  root  of  10,  for  l/2  is  not  a 
commensurable  fraction._  While  we  do  not  know  the 
meaning  or  value  of  10^^,  we  do  know  the  meaning  of 
101-^,— it  is  the  14th  power  of  the  10th  root  of  10. 
Likewise  10^*^^  has  a  meaning, — the  141th  power  ot 
the  100th  root  of  10.  It  would  involve  tedious  work  to 
compute  these  powers  and  roots  of  10,  but  if  they  be  so 
computed  we  would  find,  to  seven  places 

101-4  ^  2.511887-  .  . 

101-4  1  ^  2.570396    .  . 

101-414  ^  2.594179.  .  . 

101-4  14  2        =  2.595374.  .  . 

101.41421      ^  2.595434.  •  . 

101-414213    =2.595452... 

101-4142135  _  2.595455.  .  . 
Now  we  have  purposely  selected  closer  and  closer 
approximations  to  ]/2  as  the  exponents  of  10  in  the  left 
members  of  these  equations.  Whence  we  conclude  that 
the  right  members  of  these  same  equations  are  closer  and 
closer  approximations  to  10  ".  Thus  we  observe  that 
10  is  an  incommensurable  number  of  which  2.595455  is 
a  commensurable  approximation  to  seven  places  of 
decimals. 


LOGARITHMS.  507 

It  is  not  hard  to  see  that  we  could  get  an  approximate 
value  of  any  incommensurable  power  of  a  base  in  a  similar 
way  to  the  above.  Whence  we  make  the  following  defi- 
nition of  an  incommensurable  power  of  a  number: 

If  n  is  incommensicrable,  a''  is  the  limit  of  a"",  in  which  x 
is  an  always  cornmensurable  variable  appj'oaching  n  as  a 
limit.     In  symbols  this  statement  is  seen  to  be 

a"=limit  of  a\  or         a  "-'*°f-=  limit  of  a\         [1] 

As  a  further  illustration  of  a  number  affected  by  an 
incommensurable  exponent,  consider  the  expression  a'^ 
where  tt  represents  the  ratio  of  the  diameter  of  a  circle  to 
the  circumference.  It  can  be  shown  that  the  incommen- 
surable number  tt  can  be  expressed  as  the  limit  of  the 
following  infinite  series: 

^ — 4 4(4 44_  4 

That  is,  the  incommensurable  number  tt  is  expressed  as 
the  limit  of  the  commensurable  variable  in  the  right 
member. 

Then  we  have 

4_4    I    4_4    I    4 .    .    , 

^^=hmit  a 
Thus  a'^  is  the  limit  of  a^,  in  which  x  is  the  commen- 
surable variable  4— f  +  f— ^+  •  •  •  which  approaches  the 
incommensurable  limit  tt. 

676.  Since  in  the  expression  a^  the  :r  appears  in  such 
an  unrestricted  form  (having  just  provided  for  incom- 
mensurable values)  it  is  common  to  speak  of  the  expression 
as  an  Exponential  Function  of  x,  intending  to  call 
attention  thereby  to  the  fact  that  x  may  be  considered  a 
continuous  variable  as  in  any  ordinary  algebraic  expres- 
sion. 

If  in  the  et^uation  ci'—y, 

we  assume  x  to  pass  from  one  extreme  of  the  algebraic 
scale  to  the  other,  taking  in  succession  every  possible 


S08  UNIVERSITY   ALGEBRA. 

value,  then  we  are  able  to  give  a  meaning  to  this  equation 
in  two  variable;  because  for  every  possible  value  of  x,  ^^, 
that  is,  jj/,  has  a  definite  meaning  and  value. 

It  must  be  remembered  that  when  we  speak  of  d^  we 
mean  that  a  is  a  positive  number,  a?id  by  the  value  of  a""  we 
mean  that  one  of  its  values  which  is  positive.  Hence  in  the. 
equation  a^'—y  we  are  to  think  of  but  one  value  oi  y 
resulting  when  any  particular  value  is  assigned  to  x. 
Thus  in  10-^=j^/ we  are  to  understand  jf=  + 1^10  and 
not  jj/=--l/lO  or  any  other  possible  value  oi  y. 

Of  course  the  very  restrictions  just  mentioned  prevent 
y  from  having  a  riegative  value.  Moreover,  it  is  not  evi- 
dent that  y  can  have  every  positive  value  w^e  please. 
For  example,  it  is  not  plain  that  a  value  of  x  exists  which 
satisfies  the  equation  10^= tt.  In  general,  while  it  is  easy 
to  see  that  in  the  equation 

a''=y 
there  always  exists  a  value  oi  y  for  any  value  assigned 
to  X,  it  is  far  from  evident  that  there  exists  a  value  of  x 
corresponding  to  every   value   which   may  be   assigned 
to  y.     Whence  the  necessity  for  the  following  theorems: 

I 

677.      The  expression  a""  can  be  Tnade  to  differ  from  1 

by  less  tha7i  any  assigned  number  ij  x  be  sufficiently  in- 
creased. 

Suppose  it  be  required  to  increase  x  so  that 

a-<l  +  ^,  (1) 

in  which  d  stands  for  an  assigned  positive  number  how- 
ever small.  Then  we  must  prove  that  x  can  be  taken 
large  enough  to  give 

{\^dy>a, 
or,  by  the  binomial  theorem,  x  must  be  large  enough  to 

give  i4-;^^+f.^+i)^2+  .  .  .^^  (2) 


LOGARITHMS.  509 

It  is  plain  that  1  +  xd  can  always  be  made  greater  than 
a  by  taking  x  sufficiently  large.     In  fact,  the  inequality 

l+xd>a 
will  hold  if  ,  xd^a—1, 

or  xf  ^>—J — 

a 

Now,  since  l  +  xdy-a  if  :t:>— -7— ?  then  much  more  is 

the  left  member  of  (2)  greater  than  the  right  member  if 

Hence,  to  make  w  less  than  1  +  d  take 

a-1 
^>-d~ 

(1)     Find  X  such  that  loi<1.0001. 
Here  ^==.0001  and  ^  =  10;  whence 
9 

678.  Laws  of  Incommensurable  Indices.  In  this 
article  x  and  y  stand  for  two  commensurable  variables 
which  approach  the  incommensurable  limits  n  and  r  re- 
spectivel}^ 

Therefore  we  know,  x  and  j/  being  commensurable, 

a-a^=^"+'\  (1) 

Taking  limit  of  each  side,  Art.  420, 

limit  a-*'^^=  limit  ^-^+-^,  (2) 

and  by  Art.  424, 

limit  a"-'  limit  ^■^=  limit  «*'"^-^.  (3) 

But  by  [1]  limit  a''= « '"""''=«'',  and  similarly  for  any 
o*:her  variables  as  y  or  {x-\-y). 

Therefore,  a"a^=^"+'',  [2] 

or  tile  law  of  exponents  in  multiplication  \%  proved  for 
incommensurable  exponents. 


5IO  UNIVERSITY   ALGEBRA. 

Starting  with  a''-^a^=a''-^  it  can  be  shown  in  an  iden- 
tical way  that 

holds  for  incommensurable  exponents. 

Because  x  and  y  are  commensurable,  we  have 

{a^y=^a^\  (4) 

Whence,  by  Art.  420  and  [1], 

limit  (^")-^=  limit  a''^=a''\  (.5) 

Now  let  x—ii-^tt\  whence  21  is  a  variable  whose  limit  is 
zero.     Then  we  have 

{af'y=  (^"+'0''=  {a^'a^y^  {cCy^a'y  (6) 

Whence 

limit  (^T=  limit  [(^"^(^'0^]  =  limit  {ary  limit  {cCy.  (7) 
But  limit  ?^^0,  therefore  limit  {cCy—\  by  Art.  677. 
Hence,  comparing  (5)  and  (7),  we  have 

a""=  limit  {cCy. 
By  [1]  limit  {cCy^ia'y, 

whence  we  have  proved 

{a:y==a^^\  [4] 

for  incommensurable  exponents. 
Starting  with 

{abcy^^a'^b^c^  and  gy=  ~ 

it  is  easy  to  prove  the  formulas 

{abcy=^a''y'd\  [5] 

for  incommensurable  exponents. 

679.      The  expressio7i  a^  is  a  contijitwics  function  of  x. 

Suppcse  a''=y  and  let  x  take  on  any  increase,  ^^  and 
suppose  the  corresponding  value  of  y  be  j/+A  so  that 

a^^'=^y-^t.  (1) 

We  are  to  prove  that  as  x  passes  cofitiritionsly  from  x  to 
x-\-s  then  jK  passes  contimioiLsIy  from  jj'  toj/+/;  that  is,  as 
X  changes  from  x  X.Q  x-\-s  by  passing  over  every  iiitervie- 


LOGARITHMS.  51I 

diate  value  that  j/  changes  from  jk  to  y-\-t  by  passing  over 
every  intermediate  value. 

The  equation  a''^'=^y-\-t  may  be  written 

a^a'=.yj^t,  (2) 

and  since  a'—-y,  this  may  be  written 

a'^a'^a'-^t,  (3) 

or  a\a'—V)^t.  (4) 

Now.,  by  Art.  677,  by  taking  ,y  small  enough  <2'"maybe 
made  to  differ  from  1  by  an  amount  as  small  as  we  please. 
Hence  in  equation  (4)  t  may  be  made  as  small  as  we  please 
by  taking  ,?  small  enough.  That  is,  the  difference  between 
two  successive  values  of  ^^'^  can  be  made  as  small  as  we 
please.     Therefore  it  is  a  continuous  function  of  .r. 

680.  It  follows  directly  from  the  above  article  that 
for  every  positive  value  which  may  be  assigned  to  y  in  the 
equation  a^'^y^  a  corresponding  value  of  x  exists  which  will 
satisfy  the  equatiori. 

For  the  last  article  shows  that  if  x  is  increased  contin- 
uously from  the  value  0  without  limit  then  y  increases 
continuously  from  the  value  1  withoutlhiiit.  Thatis,  ymay 
have  every  value  greater  than  1.  It  is  also  seen  that  as 
X  is  decreased  continuously  from  the  value  0  without 
limit  that  y  decreases  continuously  from  the  value  1. 
That  is,  y  may  have  every  fractional  value. 

The  above  states  that  if  any  value  be  assigned  to  y  in 
the  equation  a*==_y  that  a  value  of  x  exists  which  will 
satisfy  it,  but  it! does  i^ot  explain  how  to  find  that  value. 
Thus  it  does  not  show  how  to  find  x  in  the  equation 
10"'=  5.  The  method  of  finding  this  will  be  explained 
later. 

I^OGARITHMS. 

681.  In  the  equation  10''=_>',  x  is  called  the  Common 
Logarithm  oiy.     That  is,  the  common  logarithm  of  any 


512  UNIVERSITY    ALGEBRA. 

number  is  the  exponent  of  the  power  to  which  10  must 
be  raised  to  produce  the  given  number. 

Thus  2  is  the  common  logarithm  of  100,  since  10- =  100  ;  likewise. 
1.301030  is  the  common  logarithm  of  20,  since  10^  •301030—20. 

682.  In  the  equation  a''=y,  in  which  a  is  a  positive 
number  not  1 : 

The  constant  a  is  called  the  Base. 

The  number  y  is  called  the  Exponential  of  x  to  the 
base  a,  and  may  be  written  jk=<3:'''  orjF=exp^.r. 

The  number  x  is  called  the  LrOgarithm  of  y  to  the 
base  a,  and  is  written  x=\og^y. 

The  use  of  the  word  logarithm  may  be  kept  in  mind  by  remem- 
bering this  sentence  :  In  the  equation  «'^=y,  x  is  called  the  Exponent 
of  the  power  of  a  or  the  Logai'itJijji  oi  y. 

Of  course  the  two  equations 

a-^y  (1) 

x=\o%ay  (2) 

express  the  same  truth  respecting  the  relation  between  x  and  y.   The 

second  equation  uses  the  logarithmic  notation  and  is  always  to  be 

interpreted  by  means  of  the  first  equation, 

683.  Systems  of  Logarithms.  If  in  the  equation 
a'^^y,  where  a  is  any  positive  constant  not  1,  different 
values  be  assigned  to  y  and  the  corresponding  values  of 
x  be  computed  and  tabulated,  the  results  constitute  a 
System  of  Logarithms. 

Since  any  positive  value  except  1  may  be  chosen  for 
the  base  a,  the  number  of  different  possible  systems  of 
logarithms  is  unlimited.  As  a  matter  of  fact,  however, 
only  two  systems  are  now  used  ;  the  Natural  or  Nape- 
rian  or  Hyperbolic  System,  whose  base  is  approxi- 
mately 2.7182818  + ,  and  the  Common  or  Briggs' 
System,  whose  base  is  10. 

The  Natural  logarithms  of  all  numbers  from  1  to  20,000  have  been 
computed  to  17  places  of  decimals.  The  common  logarithms  of  all 
numbers  from  1  to  over  200,000  have  been  found.  They  are  usually 
printed  in  tables,  to  four,  five,  six,  cr  seven  decimal  places. 


8. 

10-25  =  i.7782 

9. 

^"+^'=a''<2''. 

10. 

J^Q. 301030  —  3^ 

II. 

a^—a. 

12. 

a\ogay=y. 

13. 

101ogio/=j/. 

14. 

e^  =  a 

LOGARITHMS.  5  I  3 

In  the  following  pages  the  common  logarithm  of  any 
number,  n^  will  be  denoted  by  the  symbol  log  n,  and  not 
by  logio?2.  Thus  the  base  is  supposed  to  be  10  unless 
otherwise  indicated. 

KXAMPIyKS. 

Write  the  following  equations,  using  the  logarithmic 
notation: 

I.         10"=  TT. 

2.  e^=y. 

3.  112  =  121. 

4.  10^  =  1000. 

5.  16-25  =  2. 

6.  10^  =  1. 

7.  10-3  =  . 001. 

15.     10^ '  =  2.595455. 
Express  the  following,  using  the  exponential  notation  : 

16.  log27(i)=-.3333-f      20.     Iog2l024=10. 

17.  log  io4=.  602060.  21.     log,^=l. 

18.  log  10 10000= 4.  22.     \og,b'=b, 

19.  logi  0.00001  = -5.  23.     \og,a==B, 

24.     Iog,ol^l00=ilog,ol00=f. 

PROPKRTIKS  OF  I^OGARITHMS. 

684.  Inasmuch  as  logarithms  are  merely  the  exponents 
of  a  fixed  base,  the  properties  of  logarithms  are  entirely 
dependent  upon  the  properties  of  exponents  in  general, 
which  have  already  been  established. 

Among  the  fundamental  properties  of  logarithms  are 
these  : 

33- u.  A. 


514  UNIVERSITY    ALGEBRA.       . 

The  logarithin  of  the  base  itself  iyi  any  system  is  1. 

For  a^-=a, 

that  is,  log,,a=l. 

The  logarithm  of  uiiity  in  all  systems  is  0. 

For  a<>  =  l, 

that  is,  logj=0. 

Negative  numbers  have  no  logarithins. 

For  in  the  equation  a''=y,  a  is  positive,  and  by  the 
value  of  a""  we  mean  that  one  of  its  values  which  is  posi- 
tive, because  of  the  restrictions  we  previously  imposed. 
Hence  j^  cannot  be  negative.     See  Art.  674. 

If  we  understand  the  same  system  of  logarithms  to  be 
used  throughout,  then  the  following  four  theorems  hold: 

685.     Logarithm  of  a  Product.     I,et  7i  and  r  be  any 

two  positive  numbers  and  let 

logji=x  and  log^r=j/.  (1) 

ITow,  from  (1)  and  definition  of  a  logarithm 
n=a^  r=a^. 

Multiplying  nr=a''a''=a''+^'  by  Art.  678. 

Therefore,  by  definition  of  a  logarithm, 

logjir=x-{-y 
as  by  (1)  log.nr=log.n+log.r.  [7] 

Hence,  the  logarithni  of  the  pjvdtict  of  tivo  numbers  is 
equal  to  the  sum  of  the  logarithms  of  those  7iumbers, 
In  the  same  way,  if  log««y=<3',  then 

that  is,  \o%jirs=\oZan-\-\o%,,r-\-\o%aS 

EXAMPLES. 

1.  Given  log  2=0.3010  and  log  3=0.4771;  find  log  6. 

2.  Given  log  5=0.6990  and  log  7=0.8451 ;  find  log  35. 

3.  Given  log  12  =  1.0792  ;  find  log  144. 


LOGARITHMS.  SI  5 

4.  Given  log  739=2.8686  and  log  642=2.8075;  find 
logarithm  of  product. 

5.  Given  log  22;i;=1.9445    and  log  22  =  1.3424;    find 
log  X. 

6.  Given  log  20=1.3010;  find  log  200. 

686.     Logarithm  of  a  Quotient.      I^et  n  and  r  be 
any  two  positive  numbers,  and  let  a 

log,^7^=-r  and  log.^'^i^xy  (1) 

From  (1)  and  definition  of  a  logarithm, 
71= a""  r=a^. 

Dividing  -=^--i-a^'=^"-^  by  Art.  678. 

Therefore,  by  definition  of  a  logarithm, 

1       ^^ 

loga-==-^— Ti 
r 

or  by  (1)  log,^  =log.ri-log.r.  [8] 

Therefore,  the  logarithm  of  the  quotient  of  two  tuunhers 
equals  the  logarithm  of  the  dividend  minus  the  logarithjn  of 
the  divisor, 

KXAMPI^ES. 

1.  Given  log75=1.8751andlog  15=1.1761;  find  log 5. 

2.  Given  log  60=1.7782  and  log  12=1.0792;  find  log  5. 

3.  Given  log84=1.9243  and  log  12=1.0792;  find  log  7. 

4.  Given  log  435=2.6385  and  log  317=2.5011;  find 

43.5 
3 1 7  • 

5.  Given  log  y\x=  1.7761  and  log  6=.7782;  find  log  x. 


log  -^-^^ 


687.     Logarithm  of  a  Power.     I,et  71  be  any  posi- 
tive number,  and  let 

\o<gji=x,  (1) 


5l6  UNIVERSITY    ALGEBRA. 

From  (1)  and  definition  of  a  logarithm 
Raising  both  sides  to  the  pth  power 
Therefore,  by  definition  of  a  logarithm, 

or  by  (1)  \ogan^=p  logati.  [9] 

Therefore,  ^ke  logarithm  of  any  power  of  a  Clumber  equals 
the  logarithm  of  the  7iiimber  7nultiplied  by  the  index  of  the 
power. 

688.     Logarithm  of  a  Root.     I^et  n  be  any  positive 
number,  and  let 

\ogji=x.  (1) 

From  (1)  and  definition  of  a  logarithm 

Taking  the  ^th  root  of  both  sides 

X 

yn=a^' 
Therefore,  by  definition  of  a  logarithm, 

log^V    71=  -, 

or  by  (1)  log,#/n=  1^^.  [10] 

Therefore,  the  logarithm  of  any  root  of  a  number  equals 
the  logarithm  of  the  7iumber  divided  by  the  i7idex  of  the  root. 

EXAMPLES. 

I.  Given  log  2=.3010;  find  log  1024. 


2.  Given  log  1111  =  3.0457;  find  log  1^1111. 

Apply  as  far  as  possible  the  principles  just  proved  and 
express  the  following  logarithms  in  simplest  form : 

3.  logl^/^x^^-^  _ 

log  Vl  X  1^' ¥=iog  l^^i  +  log  f'^^ 

=ilog|  +  ,MogV- 
=  1  log  7-1  log  9  +  i  log  15-i  log  7. 
=  (i-|)  log  l-\  log  3-f-i  log  3  +  ^  log  5. 
=^'d  log  7-/o  log  3  +  1  log  5 


LOGARITHMS.  51/ 

4.  log  {hVlii'aH^').  9.  log  (1-^11+ l^S). 

5.  log  (a#'4al>'2«^).  10.  log  (I'-^ol^  +  v'^^T). 

6.  log  (3l/2^x5l'/272).  II.  logCl^aVT^^a^). 

7.  log  (f/  9-4-|/3).  12.  log  (1  ^8^1/2^). 

8.  log  1^"  12-- t/ 6).  13.  log(2lKi^-T-l'V). 

14.  loe:  — 7=.        15.  loo:  — .    .    -  .       10.  loo"       i     • 
17.  log[(^2#(^^^)*].     20.  log(^-^yY 


„     ,         /jrO'2^     2\3  10^  <! 

18.  logf--^ ^)  21.  log-— 


i  1. 


,       /;i;2      >/n  ,      VaV?xi 

19.  logf-^^^YJ  22.  log-       ,  -    3 

23.  Prove  that  loga^= 


log.« 

CHARACTERISTIC   AND   MANTISSA. 

689.  For  reasons  which  will  appear  later  the  common 
logarithm  of  a  number  is  always  written  so  that  it  shall 
consist  of  a  positive  decimal  part  less  than  1  and  an 
integral  part  which  may  be  either  positive  or  negative. 

When  a  logarithm  of  a  number  is  thus  arranged,  special 
names  are  given  to  each  part.  The  positive  or  negative 
integral  part  is  called  the  Characteristic  of  the  logar- 
ithm. The  positive  decimal  part  is  called  the  Mantissa. 
Thus,  in  log  200=2.3010,  2  is  the  characteristic  and 
.3010  the  mantissa. 


5i8  UNIVERSITY   ALGEBRA. 

690.  Since  10^  =  1  and  10^  =  1,  that  is,  since  log  1=0 
and  log  10=1,  it  follows  from  Art.  679  that  any  number 
lying  between  1  and  10  has  for  its  logarithm  a  number 
lying  between  0  and  1;  that  is,  a  proper  fraction.  Thus, 
log  2=.3010,  log  9=.9542,  log  1.56=. 1931. 

Starting  with  the  equation 

log  1.56=. 1931 
we  have 

logl5.6=log(l. 56x10)  =logl.56  +  logl0  =.1931  +  1. 
logl56.=log(l. 56x100)  =logl.56+logl00  =.1931+2, 
Iogl560=log(1.56xl000)=logl.56+log  1000=. 1931+3, 

etc.,  etc.,  etc. 
lyikewise 

log. 156  =log(l. 56-10)  =logl.56-logl0  =.1931-1, 
log.0156  =log(l. 56-100)  =logl  56-loglOO  =.1921-2, 
log.00156=log(1.56-1000)=logl.56-logl000=.1931-3, 

etc.,  etc.,  etc. 
We  observe  that  log  15.6,  log  156,  log  1560,  etc.  have 
for  their  characteristics  1,  2,  3,  etc.  respectively,  the 
mantissa  of  each  of  these  logarithms  being  .1931.  I^ike- 
wise  log  .156,  log  .0156,  log  .00156,  etc.,  have  for  their 
characteristics  —1,  —2,  —3,  etc.  respectively,  the  man- 
tissa being  .1931,  as  before.  Thus  we  see  that  the  value 
of  the  characteristics  of  these  logarithms  is  dependent 
merely  upon  the  position  of  the  decimal  point  in  the 
number. 

691.  To  Find  the  Characteristics.  1.  The  charac- 
teristic of  the  common  logarithm  oj  a?iy  number  greater 
than  unity  is  one:  i,KSS  than  the  number  of  figures  preceding 
the  decimal  point, 

2.  The  characteristic  of  the  common  logarithm  of  a  num- 
ber less  than  7mity  is  negative  and  numerically  on:^  more 
than  the  number  of  zeros  immediately  following  the  decimal 
poi7it. 


LOGARITHMS.  519 

To  prove  these  statements  we  have  merely  to  generalize 
the  illustration  given  above.  Thus,  instead  of  1.56,  let 
us  use  a  to  stand  for  any  number  between  1  and  10,  and 
let  m  be  its  logarithm.  Then,  since  log  1  =  0  and  log 
10=1,  7n  must  be  a  proper  fraction,  as  .1931  in  illustration 
above.     Then  we  have 

log  a=7n^ 
log      10^=;;/  +  1,  log      -^-^a  —  fn  —  l, 

log    100a=?;^^-2,  log    ^^-^a^m—^, 

log  1000^=;;z  +  3,  etc.  log  -^-^^-^a=m—o,  etc. 

In  this  case  the  mantissa  ot  each  logarithm  is  m,  and 
the  characteristics  are  respectively  0,  1,  2,  3,  —1,— 2,  —3. 
But,  sin^e  a  is  a  number  with  one  figure  to  the  left  of 
of  the  decimal  point,  10^  is  a  number  with  two  figures  to 
the  left  of  the  decimal  point,  100^  is  a  number  with  three 
figures  to  the  left  of  the  decimal  point,  and  so  on.  Then 
we  see 

No.  of  figures  preceding  Characteristic  of 

decimal  point  in  number.  Logarithm. 

1,  0 

2,  +1 

3,  etc.  +2,  etc. 
Thus  we  observe  that  the  characteristic  of  the  logarithm 

is  always  one  less  than  the  number  of  figures  in  the  num- 
ber preceding  the  decimal  point. 

Similarly,  since  «  is  a  number  with  one  figure  to  the 
left  of  the  decimal  point,  -^^a  is  a  number  less  than  1  with 
no  zero  immediately  following  the  decimal  point,  -^^^a 
is  a  number  with  one  zero  immediately  following  the 
decimal  point,  y-^Vir^  ^^  ^  number  with  two  zeros  im- 
mediately following  the  decimal  point,  and  so  on.  Then 
we  have 

No.  of  zeros  following  Characteristic  of 

decimal  point  in  number.  ■        Logarithm. 

0,  -1, 

1,  -2, 

2,  etc.  —3,  etc. 


520  UNIVERSITY    ALGEBRA. 

Thus  the  characteristic  of  the  logarithm  of  a  number  less 
than  unity  is  negative  and  numerically  one  more  than  the 
number  of  zeros  immediately  following  the  decimal  point. 

Negative  characteristics  are  sometimes  written  as  in 

log  .00156=3.1931,  instead  of  as  in  log  .00156=. 1931-3, 
but  preferable  to  either  of  these  is  the  notation  log  .00156 
=  7.1931  —  10,  made  by  adding  10  to  the  negative  char- 
acteristic and  then  subtracting  10  to  preserve  the  value 
unaltered. 

Some  prefer  to  use  the  following  statements  in  determining  charac- 
teristics. The  only  difference  from  the  above  being  in  counting 
from  units''  place  instead  of  from  the  decimal  point,  which  is  to  the 
right  of  units'  place. 

The  characteristic  of  the  common  logarithm  of  a  numhS  equals  the 
number  of  places  the  first  significant  figure  of  the  number  is  removed 
from  units'  place,  and  is  positive  if  the  first  significant  figure  stands 
to  the  left  of  units'  place  and  is  negative  if  it  stands  to  the  right  of  units' 
place. 

Thus,  log  1.56=0.1913,  since  first  significant  figure  is  removed  0 
places  from  units'  place;  log  1500=3.1931  since  first  significant  figure 
stands  3  places  to  the  left  of  units'  place;  and  log  .000156=171931 
since  first  significant  figure  stands  4  places  to  the  right  of  units'  place. 

KXAMPI^KS. 

1 .  What  numbers  have  0  for  the  characteristic  of  their 
logarithms?  What  numbers  have  0  for  the  mantissa 
of  their  logarithms  ? 

2.  Find  by  inspection  the  characteristics  of  the  log- 
arithms of  the  following  numbers:  5123,  647152,  41.4, 
257.752,  5,  5.5,  .5.  .0000089,  .0010089. 

3.  Given  log  1235=3.0917,  write  the  logarithms  of  the 
following  numbers:  12.35,  1235000,  1.235,  .001235, 
123.5,  .1235,  .00001235,  .01235000. 

4.  Given  log  476=2.6776,  write  down  the  numbers 
which  have  the  following  logarithms:    1.6776,  6.6776, 


LOGARITHMS.  .  521 

10.6776,  0.6776,  3.6776,  3.6776,  1,6776,  5.6776,  4.6776, 
8.6776. 

5.  Given  log  2=0.30103  and  log  3=0.47712,  find  log 
64;  also  log  182. 

6.  Show  that  log  ii+log  ffr-^  log  |=log  2. 

,      ,      62x53x(75)^x4x3     ..      ^    i      k 

7.  Showthatlog ^^3x20^ i-=i-log3-log  5. 

14^x30 

8.  Show  that  log ^^^o""""^^^^""^^^^- 

g.  Given  log  2=0.30103,  find  Jog  1/1725. 

Hint:  V1.25=V|=VJ30. 

10.  Given  log  2=0.30103,  find  log  (3.125)^ 

11.  Given  log  1331  =  3.1242  and  log  539  =  2.7316,  find 
log  7  and  log  11. 

Solution:  1331  =  11x11X11  =  1^ 

and  539r=llX7X7:=llX72. 
Hence.  log  1331=3  log  11 

and  log  539=log  11  +  2  log  7. 

Hence  we  have  3  log  11  =  3.1242 

and  log  11+2  log  7=2.7310. 

Hence,  log  11  =  1.0414. 

Hence.  2  log  7=1.6902. 

Therefore.  log  7=0.8451. 

12.  Given  log  144=2.1584  and  log  324=2.5105,  find 
log  2  and  log  3. 

13.  Find  an  expression  for  the  value  of  x  from  the 
equation  3''=567. 

Solution:  Take  the  logarithm  of  each  member  and  we  have 
X  log  3=log  567. 
But  567=7X44, 

therefore,  x  log  3=4  log  3  f-log  7. 

4  log  3  + log  7 


Hence 

i.  e.,  x=4-\- 


log3 
log  7 
log  3 


522  UNIVERSITY    ALGEBRA. 

14.  Find  an  expression  for  x  in  the  equation  B""  =405. 

15.  Find  an  expression  for  x  in  the  equation 

3"  X  2^+ 1  =  1/612. 

16.  Find  an  expression  for  x  in  the  equation 

5^+2x23  =  6"-ix2"-^i. 

17.  Given  log  2=. 30103  and  log  3=. 47712,  find  how 
many  digits  to  the  left  of  the  decimal  point  in  (f)^  ^  ^  ^. 

18.  Given  log  2=. 30103  and  log  3=. 47712,  find  how 
many  digits  to  the  left  of  the  decimal  point  in  6^^. 


19.  If  jK=10i^^^^^and<3'=10i-^«&-^show  that.r=10i-^«g  •^. 

20.  If  the  logarithms  of  a,  b,  c  be  respectively  x,  j/,  2, 
prove  that  a-^-"<^^-"<r"--^=  1 . 

21.  Prove  that  log  -^^--2  log  |+log/:f2_=log  2. 

TABI^KS  OF  I^OGARITHMS, 

692.  A  table  of  logarithms  contains  the  jnaniissas  of 
the  logarithms  of  all  numbers  from  1  to  a  certain  number. 
The  tables  on  pages  526-535  contain  the  mantissas  of  the 
logarithms  of  the  nmnbers  from  1  to  1300,  printed  to  four 
decimal  places.  Characteristics  are  never  printed  in  a 
table  of  logarithms,  since  the  characteristic  of  the  log- 
arithm of  any  number  can  always  be  found  by  inspection. 
In  order  to  save  room  the  decimal  points  which  belong 
before  the  mantissas  are  omitted. 

The  table  of  logarithms  of  numbers  consists  of  six 
pages.  The  columns  headed  "no."  contain  the  num- 
bers, and  the  columns  headed  ' '  log. ' '  contain  the  man- 
tissas. The  mantissa  of  the  logarithm  of  any  number 
will  always  be  found  immediately  to  the  right  of  that 
number.  In  the  column  headed  "d."  are  printed  the 
difference  s\i^\,v^^^n  every  two  consecutive  mantissas,  which 


LOGARITHxMS.  523 

are  called  the  Tabular  Differences.  It  will  be  noticed 
that  43  is  printed  for  .0043  (the  true  difference)  and  like- 
wise for  the  other  differences. 

If  the  tabular  differences  be  multiplied  successivel}-  by 
the  numbers  .1,  .2,  .3,  .4,  .5,  6,  .7,  .8,  .9,  the  results  are 
called  Proportional  Parts,  and  can  always  be  found  in 
column  "pp."  of  the  tables. 

It  is  a  general  principle  in  computations  of  all  kinds  that,  when 
any  of  the  last  figures  of  a  number  are  to  be  neglected,  the  last  figure 
retained  should  be  increased  by  1  if  the  neglscted  figures  make  more 
than  .5.     Thus,  _ 

1/ 2=1,41421350  + 
to  eight  decimal  places;  but  if  we  use  this  to  six  places,  we  write 

1/2  =  1.414214, 
and  to  four  places,  ]/  2=1.4142. 

If  the  figures  dropped  make  exactly  .5  it  is  indifferent,  of  course^ 
whether  we  increase  the  last  figure  or  not.  A  very  good  rule  to  follow, 
however,  is  to  always  increase  the  last  figure  zv/ien  5  ts  dropped  if  this 
makes  the  result  an  even  number,  and  not  in  other  cases.  That  is,  for 
.00215  we  would  use  .0022  to  four  places;  for  00525  we  would  use  .0052 
to  four  places.  Thus  in  a  long  piece  of  numerical  work  we  would 
probably  increase  the  last  figure  about  as  many  times  as  we  drop  the 
.5,  so  would  probably  get  a  more  accurate  result  than  if  we  always 
increased  the  last  figure. 

693.  To  Find  the  Logarithm  of  a  Number.  The 
following  examples  explain  the  manner  of  using  the  table 
of  logarithms. 

(1)  Find  the  logarithm  of  250. 

The  characteristic  of  log  250  we  know  is  2.     We  find  250  in  column 
"no,"  page  520,  and  to  right  of  this  in  column  "log,"  are  the  figures 
4082,  which  mean  that  mantissa  of  log  250  is  .4082.     Therefore, 
log  250=2.4082. 

(2)  Find  the  logarithm  of  8300. 

The  characteristic  of  th3  required  logarithm  we  find  is  3.  The 
table  only  runs  as  far  as  1300,  but  we  know  that  log  8300  will  have 
the  same  mantissa  as  log  8;)0,  which  latter  is  readily  found  to  be  .9222. 
Therefore, 

log  8330=3.9222. 


524  UNIVERSITY    ALGEBRA. 

(3)  Find  the  logarithm  of  .00341. 

We  know  the  characteristic  of  log  .00341  is  —3.  The  mantissa  is 
the  same  as  that  of  the  logarithm  of  341,  which  latter  we  find  from  the 
table  to  be  .5328.     Therefore, 

log  .00341=7.5328-10. 

(4)  Find  the  logarithm  of  8. 

The  characteristic  of  log  8  is  0.  The  table  begins  at  100,  but  the 
mantissa  of  log  8  is  the  same  as  that  of  800,  which  is  in  the  table. 
Therefore, 

log  8=0.9031. 
(5).   Find  the  logarithm  of  263.6. 
The  characteristic  is  2.     By  the  table 

log  264=2.4216 
log  263=2.4200 
Difference  between  the  logarithms=  16 

which,  as  we  have  said,  is  called  the  tabular  difference  and  is  printed 
in  small  figures  in  column  "d. "  between  the  mantissas  4200  and  4216. 
If  the  difference  for  1  unit  in  the  number  is  16  in  the  logarithm  we  assume 
that  the  difference  for  .6  of  a  unit  in  a  number  will  be  .6X16,  or  9.6 
in  the  logarithm;  where  the  decimal  point  in  9.6  separates  the  fourth 
and  fifth  places  of  decimals.     Therefore, 

log  263=2.4200 
plus  the  correction!  for  .6;  10 

gives  log  263.6=2.4210. 

The  tabular  difference  multiplied  by  any  number  of  tenths  can  be 
found  in  column  "pp."  on  page  527,  under  16;  .6X16  is  in  the  sixth 
line  under  16. 

(6)  Find  the  logarithm  of  163.8. 

We  find  log  163=2.2122.  The  tabular  difference  (from  column 
"d.")  is  26,  which  is  the  amount  the  logarithm  will  change  if  the 
number  changes  one  unit.     Therefore, 

log  163=2.2122 
plus  the  correction  for  .8,  21 

gives  log  163.8=2.2143 

(7)  Find  the  logarithm  of  1292 

Since  the  table  runs  as  far  as  1300,  this  can  be  found  directly  from 
page  531,  where  it  is  seen 

log  1292=3.1113. 

(8)  Find  the  logarithm  of  469. 75. 
The  characteristic  of  log  469. 75  is  2. 

log  469=2.6712 

plus  correction  for  .7.  6.3 

plus  correction  for  .05,  .45 

gives  log  469.75=2.6719~ 

The  correction  for  .05  is  found  by  first  finding  the  correction  for  .05, 
which  is  given  as  4.5.  Since  the  correction  for  .5  is  4.5  we  assume 
the  correction  for  .05  must  be  .45. 


-m 


LOGARITHMS.  52$ 

694.     To  Use  the  Table  of  Antilogarithms.     If  a 

logarithm  is  given  and  we  wish  to  find  the  number  which 
has  this  logarithm,  it  is  convenient  to  have  a  table  in  which 
the  logarithm  is  printed  before  the  numbers.  Such  a  table 
is  called  a  table  of  Antilogarithms.  A  table  of  this  kind 
is  given  on  pages  532,  533,  534  and  535.  In  the  column 
headed  ''log."  are  printed  all  the  mantissas  from  .000  to 
.999,  and  opposite  each  in  column  *'no."  are  the  signifi- 
cant figures  of  the  corresponding  numbers.  The  numbers 
in  columns  "d."  and  ''pp."  are  used  exactly  as  in  the 
table  of  logarithms. 

(1)  Find  the  number  whose  logarithm  is  2.4780. 

The  characteristic,  2,  will  only  affect  the  position  of  the  decimal 
point  in  the  required  number,  so  we  seek  merely  the  mantissa  .4780. 
This  is  found  on  page  533,  and  opposite  it  are  the  figures  3006,  which 
are  the  significant  figures  of  the  number  required.  The  characteristic 
being  2,  we  place  the  decimal  point  between  the  0  and  6.  Hence, 
number  whose  logarithm  is  2.4780=300.6. 

(2)  Find  the  number  whose  logarithm  is  1.3648. 

We  merely  seek  the  mantissa  from  the  table,  as  before,  finding 
Number  whose  logarithm  is  1.346.  =23.12 

plus  correction  for  .8  (where  tab.  dif.  =5)  4 

gives  number  whose  logarithm  is      1.3648=23.16 

KXAMPI.KS. 
Find  the  logarithm  of  each  of  the  following  numbers 

11.  21647. 

12.  .0186. 

13.  150.75. 

14.  .00007. 

15.  .33333. 
Find  the  number  corresponding  to  each  of  the  following 

logarithms: 

16.  .3250.  19.  3.0082.  22.  4.0831. 

17.  2.6490.  20.  8.3999-10.  23.  7.0091-10. 

18.  1.1235.  21.  9.7481-10.  24.  0.5632. 


I. 

365. 

6. 

00713. 

2. 

1222. 

7- 

6.302. 

3- 

9000. 

8. 

834.67. 

4- 

834.6. 

9- 

2i. 

5- 

34.67. 

10. 

13.315. 

LOGARITHMS     OF    NUMBERS. 


no. 


log. 


no.  log.    d.  no.  log, 


no.  log. 


100 

lOI 

1 02 
103 

104 

105 
106 

107 
108 
109 

110 
III 

112 

114 
116 

[18 
119 

120 

121 
122 

123 

124 

125 
126 

127 
128 
129 

130 

131 
132 

133 

134 
135 
136 

137 
138 
139 
140 
141 
142 
143 
144 

145 
146 

147 
148 
149 

150 


0043 
0086 
0128 

0170 
0212 
0253 

0294 
0334 
0374 
0414 

0453 
0492 

0531 
0569 
0607 
0645 

0682 
0719 
0755 
0792 
0828 
0864 
0899 

0934 
0969 
1004 

1038 
1072 
1 106 

1139 

1173 
1206 
1239 

1271 
1303 
1335 

1367 
1399 
1430 
1461 
1492 
1523 
1553 

1584 
1614 
1644 

1673 
1703 

173^ 
1761 


43 
43 
42 
42 

42 
41 
41 

40 
40 
40 
39 
39 
39 
38 

38 
38 
37 

37 
36 

37 
36 
36 
35 
35 

35 
35 
34 

34 
34 
33 
34 
33 
33 
32 

32 
32 
32 

32 
31 
31 
31 
31 
30 
31 

30 
30 
29 

30 
29 

29  , 


1959 


2014 


2041 


2068 
2095 


2304 


187 


2788 


2810 

2833 
2856 

2878 
2900 
2923 


3010 


200 

201 
202 
203 

204 
205 
206 

207 
208 
209 

210 

211 
212 
213 

214 
215 
216 

217 

218 
219 

220 

221 
222 

223 

224 

225 
226 

227 
228 
229 

230 

231 

232 
233 

234 
235 
236 

237 
238 

239 
240 
241 
242 
243 

244 
245 
246 

247 
248 

249 

250 


3010 


3032 
3054 
3075 
3096 
3118 
3139 
3160 
3181 
3201 


3222 


3243 
3263 
3284 

3304 
3324 
3345 

3365 
3385 
3404 
3424 


3444 
3464 
3483 

3502 
3522 
3541 
3560 
3579 
3598 


3617 


3636 
3655 
3674 

3692 
3711 

3729 

3747 
3766 

3784 
3802 


3820 
3838 
3856 

3874 
3892 

3909 

3927 
3945 
3962 

3979 


260 
251 

252 
253 

254 
255 
256 

257 
258 

259 

260 

261 
262 
263 

264 
265 
266 

267 
268 
269 

270 
271 

272 

273 

274 

275 
276 

277 
278 
279 

280 

281 
282 
283 

284 
285 
286 

287 
288 
289 

290 

291 
292 
293 

294 
295 
296 

297 
298 
299 

300 


3979 


4166 
4183 
4200 

4216 

4232 
4249 

4265 
4281 
4298 


4314 


4330 
4346 
4362 

4378 

4393 
4409 

4425 
4440 
4456 


4472 

4487 
4502 
4518 

4533 
4548 
4564 

4579 
4594 
4609 

4624 


4639 
4654 
4669 

4683 
4698 
4713 
4728 
4742 
±757_ 
4771 


43 

42 

4-3 

4.2 

8.6 

8.4 

12.9 

12.6 

17.2 

16.8 

21.5 

21.0 

25.8 

25.2 

30.1 

29.4 

34-4 

33.6 

38.7 

37.8 

40 

39 

4.0 

3.9 

8.0 

7.8 

12.0 

II. 7 

16.0 

15.6 

20.0 

19.5 

24.0 

23.4 

28.0 

27-3 

32.0 

31.2 

36.0 

35.1 

37 

36 

3.7 
7.4 
II. I 

3.6 

7.2 
10.8 

14.8 
18.5 

14.4 
18.0 

22.2 

21.6 

25.9 
29.6 

25.2 
28.8 

33.3 

32.4 

34 

33 

3.4 

3.3 

6.8 

6.6 

10.2 

9.9 

13.6 

13.2 

17.0 

16.5 

20.4 

19.8 

23.8 

23.1 

27.2 

26.4 

30.6 

29.7 

31 

30 

3.1 

3.0 

6.2 

6.0 

9.3 

9.0 

12.4 

12.0 

15.5 

15.0 

18.6 

18.0 

21.7 

21.0 

24.8 

24.0 

27.9 

27.0 

28 

27 

2.8 

2.7 

5-6 

8.4 

l:t 

II. 2 

10.8 

14.0 
16.8 

13.5 
16.2 

19.6 

18.9 

22.4 

21.6 

25.2 

24-3 

41 

4.1 
8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32.8 
36.9 


38 

3.8 
7.6 
II. 4 
15.2 
19.0 
22.8 
26.6 
30.4 
34.2 


35 

3.5 
7.0 
10.5 
14.0 
17.5 
21.0 
24.5 
28.0 
31.5 


32 

3.2 
6.4 
9.6 
12.8 
16.0 
19.2 
22.4 
25.6 
28.8 


29 

2.9 
5.8 
8.7 
II. 6 
14.5 
17.4 
20.3 
23.2 
26.1 


26 

2.6 
5.2 
7-8 
10.4 
13.0 
15.6 
18.2 
20.8 


I.rXiAltlTHMS     OF     NUMBERS. 


527 


log. 


log-,     d.    no.      log.     d.    no, 


log. 


4771 
4786 
4800 
4814 

4829 
4843 
4857 

4871 


4900 


4914 


4928 
4942 
4955 
4969 
4983 
4997 
501 1 
5024 
5038 


300 

301 
302 
303 

304 
305 
306 

307 
308 

309 

310 

311 

312 

313 

314 
315 
316 

317 
318 
319 
320 
321 
322 
323 

324 
325 
326 

327 
328 

329 

330 

331 

332 
333 

334 
335 
336 

337 
338 
339 
340 

341 
342 
343 

344 
345 
346 

347  5403 


348 
349 
350 


5051 

5065 

5079 
5092 

5105 
5119 
5132 

5145 
5159 
5172 


5185 
5198 
5211 
5224 

5237 
5250 
5263 

5276 
5289 
5302 

5315' 
5328 
5340 
5353 

5366 
5378 
5391 


5416 
5428 

5441 


350 

351 
352 
353 

354 
355 
356 

357 
358 
359 
360 

361 
362 

363 

364 
365 
366 

367 
368 

369 

370 

371 

372 
373 

374 
375 
376 

377 
378 
379 
380 
381 
382 
383 

384 
385 
386 

387 
388 
389 
390 

391 
392 
393 

394 
395 
396 

397 
398 
399 
400 


5441 


5453 
5465 
5478 

5490 
5502 
55H 

5527 
5539 
5551 


5563 


5575 
5587 
5599 

5611 
5623 
5635 

5647 
5658 
5670 


5682 


5694 
5705 

5717 

5729 
5740 
5752 

5763 

5775 
5786 


5798 


5809 
5821 
5832 

5843 
5855 
5866 

5877 


5911 


5922 
5933 
5944 

5955 
5966 

5977 
5988 

5999 
6010 

6021 


400 

401 
402 
403 
404 

405 
406 

407 
408 
409 

410 

411 
412 
413 
414 

415 
416 

417 
418 
419 

420 

421 
422 
423 

424 
425 
426 

427 
428 
429 

430 

431 
432 
433 

434 
435 
436 

437 
438 
439 
440 
441 
442 
443 

444 
445 
446 

447 
448 
449 
450 


6021 


6031 
6042 
6053 
6064 

6075 
6085 

6096 
6107 
6117 


6128 
6l:"3^ 
6149 
6160 

6170 
6180 
6191 

6201 
6212 
6222 


6232 


6243 

6253 
6263 

6274 
6284 
6294 

6304 
6314 
6325 


6335 


6345 
6355 
6365 

6375 
6385 
6395 
6405 

6415 
6425 


6435 


6444 

6454 
6464 

6474 
6484 
6493 

6503 
6513 
6522 

6532 


450 

451 
452 
453 

454 
455 
456 

457 
458 
459 
460 
461 
462 
463 
464 

465 
466 

467 
468 
469 

470 

471 
472 

473 

474 
475 
476 

477 
478 
479 
480 
481 
482 
483 

484 
485 
486 

487 


490 

491 
492 
493 

494 
495 
496 

497 
498 
499 
500 


6532 


6542 

6551 
6561 

6571 
6580 
6590 

6599 
6609 
6618 


6628 


6637 
6646 
6656 

6665 

6675 
6684 

6693 
6702 
6712 


6721 


6730 

6739 
6749 

6758 
6767 
6776 

6785 
6794 
6803 


6821 
6830 
6839 


6857 
6866 

6875 
6884 
6893 


6902 


091 1 
6920 
6928 

6937 
6946 

6955 
6964 
6972 
6981 
6990 


25 

24 

[23 

2.5 

2.4 

2.3 

5.0 

4.8 

4.6 

7.5 

7.2 

b.q 

10. 0 

9.6 

9.2 

12.5 

12.0 

II. 5 

15.0 

14.4 

13.8 

17.5 

16.8 

16. 1 

20.0 

19.2 

18.4 

22.5 

21.6 

20.7 

22 

21 

20 

2.2 

2.1 

2.0 

4.4 
6.6 
8.8 

4.2 
6.3 

8.4 

4.0 
6.0 

8.0 

II. 0 

10.5 

10. 0 

13.2 

12.6 

12.0 

15.4 
17.6 
19.8 

18.9 

14.0 
16.0 
18.0 

1   ^^ 

18 

1.9 

1.8 

3.8 

3.6 

5.7 

5.4 

7.6 

7.2 

9.5 

9.0 

II. 4 

10.8 

13.3 

12.6 

15.2 

14.4 

17. 1 J 

16.2 

16 

15 

1.6 

1.5 

3.2 

3.0 

4.8 

4.5 

6.4 

6.0 

8.0 

7.5 

9.6 

9.0 

II. 2 

10.5 

12.8 

12.0 

14.4 

13.5 

17 

1.7 
3.4 
5.1 
6.8 
8.5 
10.2 

II. 9 
13.6 
15.3 


14 

1.4 
2.8 
4.2 
5.6 
7.0 
8.4 
9.8 
II. 2 
12.6 


13 

12 

11 

1.3 

1.2 

I.I 

2.6 

2.4 

2.2 

3.9 

3.6 

3.3 

5.2 

4.8 

4.4 

6.5 

6.0 

5.5 

7.8 

7.2 

6.6 

9.1 

8.4 

7-7 

10.4 

9.6 

8.8 

II. 7  J 

10.8 

9.9 

10 

9 

8 

1,0 

0.9 

0.8 

2.0 

1.8 

1.6 

3.0 

2.7 

2.4 

4.0 

3.6 

3.2 

5.0 

4-5 

4.0 

6.0 

5.4 

4.8 

7.0 

6.3 

5.6 

8.0 

7.2 

6.4 

9.0 

8.1 

7.2 

I.O(.AKITHMS     OF     M  IMHKKS. 


iU>. 

1  i^>j,- 

«i 

ao. 

log. 

d. 

110. 

log. 

d. 

no. 

log. 

d. 

pp. 

600 

501 
502 
503 

6990 

8 
9 
9 
8 

550 

551 

552 

553 

7404 
7412 

7419 

7427 

8 

7 
8 
8 

600 

601 
602 
603 

7782 

7789 
7796 
7803 

7 
7 
7 

7 

650 

651 
652 
653 

8129 

"873-6 
8142 
8149 

7 
6 

7 

7 

6998 
7007 
7016 

504 

505 
506 

7024 

7033 
7042 

9 
9 

8 

554 
556 

7435 
7443 
7451 

8 
8 
8 

604 
605 
606 

7810 
7818 
7825 

8 

7 
7 

654 
655 
656 

8156 
8162 
8169 

6 

7 
7 

507 
508 
509 
610 

511 
512 

513 

7050 

7059 
7067 

9 

8 

9 

8 

9 

8 
q 

557 
558 
559 
560 
561 
562 
563 

7459 
7466 

7474 

7 
8 
8 
8 

7 
8 
8 

607 
608 
609 

610 

611 
612 
613 

7832 
7839 
7846 

7 
7 
7 
7 
8 

7 

7 

657 
658 
659 

660 

661 
662 
663 

8176 
8182 
8189 

8195 

6 
7 
6 

7 
7 
6 

7 

7076 
7084 

7093 
7101 

7482 
7490 
7497 
7505 

7853 

I 
2 
3 
4 

y 

0.9 
1.8 
2.7 

3-6 

S 

0.8 
1.6 

2.4 

3-2 

7860 
7868 
7875 

8202 
8209 
8215 

514 
515 
516 

7110 
7118 
7126 

8 
8 
Q 

564 

565 
566 

7513 
7520 
7528 

7 
8 
8 

614 
615 
616 

7882 
7889 
7896 

7 
7 

7 

664 
665 
666 

8222 
8228 
8235 

6 
7 
6 

7 
8 
9 

4-5 
5-4 
6.3 

7.2 

517 
518 

519 

620 

521 
522 
523 

7135 
7143 
7152 

8 
9 
8 
8 

9 

8 
8 

567 
568 

569 

570 

571 

572 
573 

7536 
7543 
7551 

7 
8 
8 

7 
8 
8 

7 

617 
618 
619 

620 

621 
622 

623 

7903 
7910 
7917 
7924 

7 
7 
7 
7 
7 
7 
7 

667 
668 
669 

670 

671 
672 
673 

8241 
8248 

8254 

7 
6 

7 
6 

7 
6 

7 

7160 

7559 

8261 

7168 
7177 
7185 

7566 

7574 
7582 

7931 
7938 

7945 

8267 

8274 
8280 

524 

525 
526 

7193 
7202 
7210 

9 

8 
8 

574 
575 
576 

7589 
7597 
7604 

8 

7 
8 

624 
625 
626 

7952 
7959 
7966 

7 
7 
7 

674 
675 
676 

8287 
8293 
8299 

6 
6 

7 

527 
528 

529 

630 

531 
532 
533 

7218 
7226 

7235 

8 
9 

8 
8 
8 
8 
8 

577 
578 
579 
580 

581 
582 
583 

7612 
7619 
7627 

7 
8 

7 
8 

7 
8 

7 

627 
628 
629 

630 

631 
632 
^33 

7973 
7980 
7987 

7 
7 
6 

7 
7 
7 
7 

677 
678 
679 

680 

681 
682 
683 

8306 
8312 

8319 

^25_ 

8331 
8338 
8344 

6 
7 
6 
6 

7 
6 

7 

7243 

7634 
7642 
7649 
7657 

7993 

7251 

7259 
7267 

8000 
8007 
8014 

534 
535 
536 

7275 
7284 
7292 

9 
8 

8 

584 
585 
586 

7664 
7672 
7679 

8 

7 

7 

634 
635 
636 

8021 
8028 
8035 

7 
7 
6 

684 
685 
686 

8351 
8357 
8363 

6 
6 

7 

I 
2 

7 

0.7 
1.4 

6 

0.6 
1.2 

537 
538 
539 
540 

541 
542 
543 

7300 
7308 
7316 

8 
8 
8 
8 
8 
8 
8 

587 
588 

589 

690 

591 
592 
593 

7686 
7694 
_77oi 
7709 

8 
7 
8 

7 
7 
8 

637 
638 

639 

640 

641 
642 
643 

8041 
8048 
8055 

7 
7 
7 
7 
6 
7 
7 

7 
6 

7 

7 
6 

687 
688 
689 

690 

691 
692 
693 

8370 
8376 
8382 
8388 

6 
6 
6 

7 
6 
6 

7 

3 
4 
5 
6 

9 

2.1 

2.8 
3.5 
4.2 

4-9 

1.8 
2.4 

1.6 

n 

5.4 

7324 

8062 

"806^ 

8075 
8082 

7332 
7340 
7348 

7716 
7723 
7731 

8395 
8401 
8407 

544 
545 
546 

7356 
7364 
7372 

8 
8 

8 

594 
596 

7738 
7745 
7752 

7 
7 

R 

644 

645 
646 

8089 
8096 
8102 

694 
695 
696 

8414 
8420 
8426 

6 
6 

6 

548 
549 

7380 
7388 
7396 

8 
8 

597 
598 
599 

7760 
7767 
7774 

7 
7 

647 
648 
649 

8109 
8116 
8122 

697 
698 
699 

8432 
8439 
8445 

7 
6 

650 

7404 

8 

600 

7782 

660 

8129 

/ 

700 

8451 

0 

]LOGARITHMS     OF    NUMBERS. 


no. 

log. 

d. 

no. 

log. 

d. 

uo. 

log. 

d. 

no. 

log. 

d. 

pp. 

700 

701 
702 
703 

8451 

8457 
8463 
8470 

6 
6 
7 
6 

760 

751 

752 
753 

8751 

5 
6 
6 
6 

800 

801 
802 
803 

9031 

5 
6 
5 
6 

850 

851 
852 
853 

_9?94 
9299 
9304 
9309 

5 
5 
5 
6 

8756 
8762 
8768 

9036 
9042 
9047 

7 

704 

705 
706 

8476 
8482 
8488 

6 
6 
6 

754 
755 
756 

8774 
8779 
8785 

5 
6 
6 

804 
805 
806 

9053 
9058 
9063 

5 
5 
6 

854 
855 
856 

9315 
9320 

9325 

5 
5 

■1 

I 
2 
3 
4 

I 

7 
8 
9 

0.7 
1.4 
2.1 
2  8 

707 
708 
709 

710 
711 
712 

713 

8494 
8500 
8506 

6 
6 

7 
6 
6 
6 
6 

757 
758 
759 
760 
761 
762 
763 

8791 

8797 

8802 
8808 
8814 
8820 
8825 

6 
5 
6 
6 
6 
5 
6 

807 
808 
809 

810 

811 
812 
813 

9069 
9074 
9079 
9085 

5 
5 
6 

5 
6 
5 

857 
858 

859 
860 
861 
862 
863 

9330 
9335 
9340 

9345 
9350 
9355 
9360 

5 
5 
5 
5 
5 
5 
5 

3.5 
4.2 
4.9 
5.0 
6.3 

8513 

8519 
8525 
8531 

9090 
9096 
9IOI 

714 
716 

8537 
8543 
8549 

6 

6 
6 

764 
765 
766 

8831 

8837 

8842 

6 

5 
6 

814 

815 
816 

9106 
9II2 

9II7 

6 

5 

864 
865 
866 

9365 
9370 
9375 

5 

5 
5 

6 

717 
718 
719 

720 

721 

722 

723 

8555 
8561 

8567 

6 
6 
6 
6 
6 
6 
6 

767 
768 
769 

770 

771 
772 
773 

8848 
8854 
8859 
8865 

6 
5 
6 
6 

5 
6 
5 

817 
818 
819 

820 

821 
822 
823 

9122 
9128 
9133 
9138 

9143 
9149 

9154 

6 
5 
5 
5 
6 
5 

867 
868 
869 

870 

871 
872 
873 

9380 
9385 
9390 

5 
5 
5 
5 
5 
5 
5 

I 
2 
3 
4 

I 

9 

0.6 
1.2 
1.8 
2.4 

3.6 

4.8 
5-4 

8573 

9395 
9400 

9405 
9410 

8579 
8585 
8591 

8871 
8876 
8882 

724 
725 
726 

8597 

8603 
8609 

6 
6 
6 

774 
775 
776 

8887 
8893 
8899 

6 
6 

5 

824 
825 
826 

9159 
9165 
9170 

6 

5 

874 
875 
876 

9415 
9420 

9425 

5 
5 

5 

727 
728 
729 

730 

731 

732 
733 

8615 

8621 
8627 

8633 

6 
6 
6 
6 
6 
6 
6 

m 
778 
779 

780 

781 
782 
783 

8904 
8910 

8915 

6 
5 
6 
6 

5 
6 
5 

827 
828 
829 

830 

831 

832 
833 

9175 

9180 

9186 
9I9I 
9196 

9201 
9206 

5 
6 

5 
5 
5 
5 
6 

877 
878 

879 

880 

881 
882 
883 

9430 
9435 
9440 

9445 
9450 

9455 
9460 

5 
5 
5 
5 
5 
5 
5 

1 
2 
3 
4 
5 
6 

5 

0.5 

I.O 

1.5 
2.0 
2.5 
3-0 

3-5 

8921 

8639 
8645 
8651 

8927 
8932 
8938 

734 
735 
736 

8657 
8663 
8669 

6 
6 
6 

784 
785 
786 

8943 
8949 
8954 

6 

5 
6 

834 
835 
836 

9212 

9217 

9222 

5 
5 
5 

884 
885 
886 

9465 
9469 

9474 

4 
5 

9 

4.0 
4.5 

737 
738 
739 
740 

741 
742 

743 

8675 

8681 

8686 
8692 

6 
5 
6 
6 
6 
6 
6 

787 
788 
789 

790 

791 
792 
793 

8960 

8965 
8971 
8976 

5 
6 

5 
6 

5 
6 
5 
6 

5 
6 

837 
838 

839 

840 

841 
842 
843 

9227 
9232 
9238 

9243 
9248 

9253 
9258 

5 
6 

5 
5 
5 
5 
5 

6 

5 
5 

887 
888 
889 

890 

891 
892 

893 

9479 
9484 
9489 

5 
5 
5 
5 
5 
5 

9494 

9499' 
9504 
9509 

1 
2 
3 

4 

It 

1.2 

8698 
8704 
8710 

8982 
8987 
8993 

744 
745 
746 

8716 
8722 
8727 

6 

5 

794 
795 
796 

8998 
9004 
9009 

844 
845 
846 

9263 
9269 
9274 

894 
895 
896 

9513 
9518 

9523 

5 
5 

K 

4 

I 

I 

1.6 
2.0 

li 

3.2 

747 
748 
749 
750 

8733 
8739 
8745 
8751 

6 
6 
6 

797 
798 

799 

800 

9015 

9020 

9025 

5 
5 
6 

847 
848 

849 

850 

9279 
9284 
9289 

5 
5 
5 

897 
898 
899 

900 

9528 
9533 
9538 

5 
5 
4 

9 

U 

9031 

9294 

9542 

( 

/530 


LOGARITHMS    OF    NUMBERS. 


no. 

log. 

d. 

no. 

log. 

d. 

no. 

log. 

d. 

no. 

log. 

d. 

pp. 

900 
901 

9542 

5 
5 
5 
5 

950 

951 

9777 

1000 

lOOI 

0000 

1050 
105 1 

0212 

9547 

9782 

0004 

0216 

902 

9552 

952 

9786 

1002 

0009 

1052 

0220 

903 

9557 

953 

9791 

1003 

0013 

1053 

0224 

904 

9562 

954 

9795 

1004 

0017 

1054 

0228 

905 

9566 

4 

955 

9800 

1005 

0022 

1055 

0233 

906 

9571 

5 
5 

956 

9805 

1006 

0026 

1056 

0237 

907 

9576 

957 

9809 

1007 

0030 

1057 

0241 

908 

9581 

5 

958 

9814 

1008 

0035 

1058 

0245 

909 
910 
911 

9586 

5 
4 
5 

959 
980 
961 

9818 

1009 
1010 

lOII 

0039 

1059 
1060 
1061 

0^49^ 
0253 

9590 

9823 

0043 

9595 

9827 

0048 

d257 

5 

912 

9600 

5 

962 

9832 

1012 

0052 

1062 

0261 

913 

9605 

5 
4 

963 

9836 

1013 

0056 

1063 

0265 

z 

2 

0.5 

I.O 

914 

9609 

964 

9841 

1014 

0060 

1064 

0269 

3 

1-5 

915 

9614 

5 

965 

9845 

1015 

0065 

1065 

0273 

4 
5 

2.5 

916 

9619 

5 

5 

966 

9850 

1016 

0069 

1066 

0278 

6 
7 

3-0 
3-5 

917 

9624 

967 

9854 

1017 

0073 

1067 

0282 

8 

4.0 

918 

9628 

4 

968 

9859 

1018 

0077 

1068 

0286 

9 

4.5 

919 
920 
921 

9633 

5 
5 
5 

969 
970 

971 

9863 

1019 
1020 
1021 

0082 

1069 
1070 
1071 

0290 
0294 
0298 

9638 

9868 

0086 
0090 

9643 

9872 

922 

9647 

4 

972 

9877 

1022 

0095 

1072 

0302 

923 

9652 

5 
5 

973 

9881 

1023 

0099 

1073 

0306 

924 

9657 

974 

9886 

1024 

0103 

1074 

0310 

925 

9661 

4 

975 

9890 

1025 

0107 

1075 

0314 

926 

9666 

5 
5 

976 

9894 

1026 

01 1 1 

1076 

0318 

927 

9671 

977 

9899 

1027 

0II6 

1077 

0322 

928 

9675 

4 

978 

9903 

1028 

0120 

1078 

0326 

929 
930 
931 

9680 

5 
5 
4 

979 
980 

981 

9908 

1029 
1030 
103 1 

0124 

1079 
1080 
io8i 

0330 
0334 

9685 

9912 

0128 

9689 

9917 

0133 

0338 

932 

9694 

5 

982 

9921 

1032 

0137 

1082 

0342 

933 

9699 

5 
4 

983 

9926 

1033 

0I4I 

1083 

0346 

^ 

934 

9703 

984 

9930 

1034 

0145 

1084 

0350 

I 

Yt 

935 

9708 

5 

985 

9934 

1035 

0149 

1085 

0354 

3 

1.2 

936 

9713 

5 

4 

986 

9939 

1036 

0154 

1086 

0358 

4 
5 

1.6 
2.0 

937 

9717 

987 

9943 

1037 

0158 

1087 

0362 

6 

2.4 

938 

9722 

5 

988 

9948 

1038 

0162 

1088 

0366 

9 

2.8 

939 
940 
941 

9727 

5 
4 
5 

989 
990 
991 

9952 

. 

1039 
1040 
1 041 

0166 

1089 
1090 
1091 

0370 

3.2 
3.6 

9731 

9956 

0170 

0374 
0378 

9736 

9961 

0175 

942 

9741 

5 

992 

9965 

^ 

1042 

0179 

1092 

0382 

943 

9745 

4 
5 

993 

9969 

1043 

0183 

1093 

0386 

944 

9750 

994 

9974 

1044 

0187 

1094 

0390 

945 

9754 

4 

995 

9978 

1045 

0I9I 

1095 

0394 

946 

9759 

5 

996 

9983 

1046 

0195 

1096 

0398 

947 

9763 

997 

9987 

1047 

0199 

1097 

0402 

948 

9768 

5 

998 

9991 

1048 

0204 

1098 

0406 

949 
950 

9773 

5 
4 

999 
1000 

9996 

1049 

1050 

0208 

1099 
1100 

0410 

9777 

0000 

0212 

0414 

LOGARITHMS    OF    NUMBERS. 


631 


log. 


log.     d. 


logr. 


log. 


pp. 


1100 

IIOI 

1 02 
1 103 

104 

1 106 

1 107 
108 

1 109 

1110 
[III 

[112 
[II3 

[II4 

[II5 

[16 

[I18 
[II9 

1120 

1121 

122 

[123 

1 124 

125 

1126 

1127 

128 

[129 

1130 
1131 

1135 
136 

1137 
1138 

1139 

1140 

1 141 

1 142 

143 

[144 

[145 

146 

[147 
[148 
[149 

1150 


0414 


0418 
0422 
0426 

0430 

0434 
0438 

0441 

0445 
0449 


0453 


0457 
0461 
0465 

0469 
0473 
0477 
0481 
0484 


0492 
0496 
0500 
0504 

0508 
0512 
0515 

0519 
0523 
0522 
0531 


0535 
0538 
0542 

0546 
0550 
0554 

0558 
0561 
0565^ 
0569 

0573 
0577 
0580 

0584 
0588 
0592 

0596 
0599 
0603 
0607 


1150 

151 
152 

153 

154 
155 
156 

157 
158 
159 
1160 
161 
162 
163 

164 
165 
166 

167 
168 
169 
1170 
171 
172 
173 

174 
175 
176 

177 

178 
179 

1180 
181 
182 

183 
184 
185 
186 

187 
188 
189 

1190 
191 

192 

193 

194 

195 
196 

197 
198 
199 

1200 


0607 


061 1 
0615 
0618 

0622 
0626 
0630 

0633 
0637 
0641 


0645 


0648 
0652 
0656 

0660 
0663 
0667 

0671 
0674 
0678 


0682 


0686 


0693 

0697 
0700 
0704 

0708 
07 1 1 
0715 


0719 


0722 
0726 
0730 

0734 
0737 
0741 

0745 
0748 
07^2 

0751 

0759 
0763 
0766 

0770 
0774 
0777 
0781 
0785 
0788 
0792 


1200 

1201 
1202 
1203 

1204 
1205 
1206 

1207 
1208 
1209 

1210 
1211 
1212 
1213 

1214 
1215 
1216 

1217 
1218 
1219 

1220 
1221 
1222 
1223 

1224 
1225 
1226 

1227 
1228 
1229 

1230 
1231 
1232 
1233 

1234 
1235 
1236 

1237 
1238 
1239 
1240 
1241 
1242 
1243 
1244 

1245 
1246 

1247 
1248 
1249 

1250 


0792 

0797 
0799 
0803 

0806 
0810 
0813 

0817 
0821 
0824 


0828 


0831 

0835 
0839 

0842 
0846 
0849 

0853 
0856 
0860 
0864 


0867 
0871 
0874 

0878 
0881 
0885 


0S92 
0896 


0899 
0903 
0906 
0910 

0913 
0917 
0920 

0924 
0927 
0931 

^± 
0938 
0941 
0945 

0948 
0952 
0955 

0959 
0962 
0966 
-0969 


1250 
1251 
1252 
1253 

1254 
1255 
1256 

1257 
1258 

1259 

1260 

1261 
1262 
1263 

1264 
1265 
1266 

1267 
1268 
1269 

1270 
1271 
1272 
1273 

1274 

1275 
1276 

1277 
1278 
1279 

1280 

1281 
1282 
1283 

1284 
1285 
1286 

1287 
1288 
1289 

1290 
1291 
1292 
1293 

1294 

1295 
1296 

1297 
1298 
1299 

1300 


0969 


0973 
0976 
0980 

0983 


0990 

0993 
0997 
000 
004 
007 
on 
014 

017 
021 
024 

028 
031 
035 


038 


041 

045 
048 

052 

055 
059 

062 
065 
069 


072 


075 
079 
082 

086 


092 

096 
099 
103 
106 
109 

"3 
116 

119 
123 
126 

129 

133 
_i36 

139 


0.4 
0.8 
1.2 
1.6 
2.0 
2.4 
2.8 
3-2 

3.6 


0.3 
0.6 
0.9 

1.2 
1.5 
1.8 

2.1 
2.4 
2.7 


5352 


ANTII.OGA  RITHMS. 


log:. 

no. 

d. 

log. 

no. 

d. 

log:. 

no. 

d. 

log. 

no. 

d. 

log. 

no. 

d. 

pp. 

.000 
.001 

1000 

.050 

.051 

II22' 

3 
2 

.100 

.  lOI 

^259^ 
1262 

3 
3 
3 
3 

.160 

.151 

I413 

3 

.200 

.201 

1585 

4 
3 

1002 

^ 

1 125 

1416 

1589 

.002 

1005 

3 

.052 

II27 

3 
2 

.102 

1265 

.152 

I419 

3 

.202 

1592 

.003 

1007 

2 

.055 

1 130 

.103 

1268 

.153 

1422 

3 
4 

.203 

1596 

.004 

1009 

.054 

II32 

3 

.  104 

1271 

3 

.154 

1426 

3 

.204 

1600 

.005 

IOI2 

3 

.055 

II35 

.105 

1274 

•  155 

1429 

.205 

1603 

. 

.006 

IOI4 

2 
2 

.056 

II38 

3 
2 

.106 

1276 

3 

.156 

1432 

3 
3 

.206 

1607 

2 

.007 

IOI6 

.057 

1 140 

.107 

1279 

.157 

1435 

.207 

161I 

I 

0 

.og8 

IOI9 

3 

.058 

1 143 

3 

.108 

1282 

3 

.158 

1439 

4 

.208 

1614 

2 

0 

.009 
.010 
.oil 

102 1 

2 
3 

.059 
.060 
.061 

1 146 

3 

2 

3 

.  109 
.110 
.III 

1285 

3 
3 
3 

.159 
.160 

.161 

1442 

3 
3 

4 

1 

.209 
.210 
.211 

1618 
1622 

3 
4 
5 
6 
7 

1023 

1 148 

1288 

1445 

1026 

II51 

1291 

1449 

T6^ 

.012 

1028 

2 

.062 

I153 

2 

.112 

1294 

3 

.162 

1452 

3 

.212 

1629 

8 

2 

.013 

1030 

2 

3 

.063 

I156 

3 
3 

.113 

1297 

3 
3 

163 

1455 

3 
4 

.213 

1633 

9 

2 

.014 

1033 

.064 

II59 

.114 

1300 

.164 

1459 

.214 

1637 

.015 

1035 

^ 

.065 

I161 

.115 

1303 

3 

.165 

1462 

3 

•  215 

1 641 

.016 

1038 

3 
2 

.066 

1 164 

3 
3 

.116 

1306 

3 
3 

.166 

1466 

4 
3 

.216 

1644 

.017 

1040 

.067 

I167 

.117 

1309 

.167 

1469 

,217 

1648 

.018 

1042 

^ 

.068 

1 169 

^ 

.118 

1312 

3 

.168 

1472 

3 

.218 

1652 

.019 
.020 

1045 
1047 

3 

2 

.069 
.070 

II72 

3 
3 

.119 
.120 

1315 

3 
3 
3 

.169 
.170 

1476 

4 
3 

.219 
.220 

1656 
1660 

I175 

1318 

1479 

.021 

1050 

3 

.071 

I178 

3 

.121 

1321 

.171 

1483 

4 

.221 

1663 

.022 

1052 

^ 

.072 

1 180 

.122 

1324 

3 

.172 

i486 

3 

.222 

1667 

3 

.023 

1054 

3 

.073 

1 183 

3 

3 

.123 

1327 

3 
3 

•  ^73 

1489 

3 
4 

.223 

I67I 

I 

0 

.024 

1057 

.074 

1 186 

.124 

1330 

.174 

1493 

.224 

1675 

2 

3 
4 

I 

.025 

1059 

^ 

.075 

1 189 

3 

.125 

1334 

4 

.175 

1496 

3 

.225 

1679 

I 

.026 

1062 

3 

2 

.076 

II91 

2 

3 

.126 

1337 

3 
3 

.176 

1500 

4 
3 

.226 

1683 

5 
6 

2 
2 

.027 

1064 

.077 

II94 

.127 

1340 

.177 

1503 

.227 

1687 

7 

2 

.028 

1067 

3 

.078 

II97 

3 

.128 

1343 

3 

.178 

1507 

4 

.228 

1690 

8 

2 

.029 
.030 
.031 

1069 

2 

3 

2 

.079 
.080 

.081 

1 199 
1202 

2 
3 
3 

.129 
.130 
.131 

1346 

3 
3 
3 

.179 
.180 
.181 

1510 

3 
4 
3 

.229 
.230 
.231 

1694 

9 

3 

1072 

1349 

1514 

1698 

1074 

1205 

1352 

1517 

1702 

.032 

1076 

2 

.082 

1208 

3 

.132 

1355 

3 

.182 

1521 

4 

.232 

1706 

.033 

1079 

3 
2 

.083 

I2II 

3 
2 

.133 

1358 

3 
3 

.183 

1524 

3 
4 

.233 

I7I0 

.034 

I081 

.084 

I213 

.134 

1361 

.184 

1528 

.234 

I7I4 

.035 

1084 

3 

.085 

I216 

3 

.135 

1365 

4 

.185 

1531 

3 

.235 

I7I8 

.036 

1086 

3 

.086 

I219 

3 

3 

.136 

1368 

3 
3 

.186 

1535 

4 
3 

.236 

1722 

.037 

1089 

.087 

1222 

.137 

1371 

.187 

1538 

.237 

1726 

.038 

IO9I 

2 

.088 

1225 

3 

.138 

1374 

3 

.188 

1542, 

4 

.238 

1730 

4 

.039 
.040 

1094 

3 
2 

.089 
.090 

1227 

3 

.139 
.140 

1377 

3 
3 

.189 
.190 

1545 

3 
4 

.239 
.240 

1734 

I 
2 
3 
4 

0 

I 

1096 

1230 

1380 

1549 

1738 

.041 

1099 

3 

.091 

1233 

3 

.i4i 

1384 

4 

.191 

1552 

3 

.241 

1742 

2 

.042 

I  102 

3 

.092 

1236 

3 

.142 

1387 

3 

.192. 

1556 

4 

.242 

1746 

5 

2 

.043 

1 104 

2 
3 

.093 

1239 

3 
3 

.143 

1390 

3 
3 

.193 

1560 

4 
3 

.243 

1750 

3 

.044 

1 107 

.094 

1242 

.144 

1393 

.194 

1563 

.244 

1754 

9 

3 

.045 

1 109 

2 

.095 

1245 

3 

.145 

1396 

3 

.195 

1567 

4 

.245 

1758 

.046 

III2 

3 
2 

.096 

1247 

2 

3 

.146 

1400 

4 
3 

.196 

1570 

3 
4 

.246 

1762 

.047 

III4 

.097 

1250 

.147 

1403 

.197 

1574 

.247 

1766 

.048 

in7 

3 

.098 

1253 

3 

.148 

1406 

3 

.198 

1578 

4 

.248 

1770 

.049 
.060 

1119 

2 

.099 
.100 

1256 

3 

3 

.149 
.150 

1409 
1413 

3 
4 

.199 
.200 

1581 

3 
4 

.249 
.250 

1774 
1778 

1122  , 

3 

1259 

1585 

I 

ANTILOGARITHMS. 


533 


log. 

no. 

d. 

log. 

no. 

d. 

log. 

no. 

d. 

log. 

no. 

d. 

log. 

no. 

d. 

pp. 

.260 
.251 
.252 
.253 

1778 

4 
4 
5 
4 

.300 
.301 
.302 
.303 

1995 

5 
4 
5 
S 

.350 

.351 

.352 
.353 

2239 

5 
5 
5 

.400 
.401 
.402 
.403 

2512 
2518 
2523 
2529 

6 

5 
6 
6 

.450 

.451 
.452 

.453 

2818 

-  7 
6 
7 
6 

1782 
1786 
1791 

2000 
2004 
2009 

2244 
2249 
2254 

2825 
2831 
2838 

T 

4 

0 

.254 
.255 
.256 

1795 
1799 
1803 

4 
4 

4 

.304 
.305 
.306 

2014 
2018. 
2023 

4 
5 
5 

.354 
.355 
.356 

2259 
2265 
2270 

6 

5 
5 

.404 
.405 
.406 

2535 
2541 
2547 

6 
6 
6 

.454 
•455 
.456 

2844 
2851 
2858 

7 
7 
6 

2 

3 
4 
5 

I 

T 
2 
2 

.257 
.258 
.259 

.260 
.261 
.262 
.263 

1807 
181I 
1816 

4 
5 
4 
4 
4 
4 
■i 

.307 
.308 

.309 

.310 

•  311 
.312 

.313 

2028 
2032 
2037 

4 

5 
5 
4 
5 
5 

.357 
.358 
.359 
.360 

.361 
.362 

.363 

2275 
2280 
2286 

5 
6 

5 
5 
5 
6 

.407 
.408 

.409 

.410 

.411 
.412 
.413 

2553 
2559 
2564 

6 

5 
6 
6 
6 
6 
6 

.457 
.458 
•459 
.460 

.461 
.462 
.463 

2864 
2871 

2877 

7 
6 

7 
7 
6 
7 
7 

0 

7 
8 

9 

2 
3 
3 
4 

1820 

782^ 

1828 
1832 

2042 
2046 
2051 
2056 

2291 

2570 
2576 
2582 
2588 

2884 

1 

2296 
2301 
2307 

2891 

2897 
2904 

1 

5 
0 

.264 
.265 
.266 

1837 

I84I 

1845 

4 
4 
4 

.314 
.315 
.316 

2061 
2065 
2070 

4 

5 

5 

.364 
.365 
.366 

2312 

2317 
2323 

5 
6 

5 

.414 
.415 
.416 

2594 
2600 
2606 

6 
6 
6 

.464 
.465 
.466 

2911 
2917 
2924 

6 
7 

7 

2 
3 
4 

i 

z 
2 
2 
2 

.267 
.268 
.269 
.270 
.271 
.272 
.273 

1849 
1854 
1858 

5 
4 
4 
4 
5 
4 
4 

.317 
.318 
.319 
.320 

.321 
.322 
.323 

2075 
2080 
2084 

5 
4 
5 
5 
5 
5 
5 

.367 
.368 

.369 

.370 

.371 
.372 
.373 

2328 
2333 
2339 
2344 
2350 
2355 
2360 

5 
6 

5 
6 

5 
5 
6 

.417 
.418 

.419 
.420 
.421 
.422 
.423 

2612 
2618 
2624 
2630 

6 
6 
6 
6 
6 
7 
6 

.467 
.468 
.469 
.470 

•  471 
.472 
.473 

2931 
2938 
2944 

7 
6 

7 
7 
7 
7 

7 

I 
9 

4 
4 
4 

1862 

2089 

2951 

1 

1866 
I87I 

1875 

2094 
2099 
2104 

2636 
2642 
2649 

2958 
2965 
2972 

1 

6 

I 

.274 

.275 
.276 

1879 

1884 
1888 

5 
4 
4 

.324 
.325 
.326 

2109 
2113 
2118 

4 

5 

.374 
.375 
.376 

2366 
2371 
2377 

5 
6 

.424 
.425 
.426 

2655 
2661 
2667 

6 
6 
6 

.474 
.476 

2979 
2985 
2992 

6 
7 

7 

2 
3 

4 

I 
2 
2 
3 
4 

.277 
.278 
.279 

.280 
.281 
.282 
.283 

1892 
1897 

I90I 

5 
4 
4 
5 
4 
5 
4 

.327 
.328 
.329 
.330 

.331 
.332 
.333 

2123 
2128 
2133 

5 
5 
5 
5 
5 
5 
5 

.377 
.378 
.379 
.380 

.381 
.382 
.383 

2382 
2388 
2393 

6 
5 
6 

5 
6 

5 
6 

.427 
.428 

.429 

.430 

•431 
.432 
.433 

2673 
2679 
2685 

6 
6 
7 
6 
6 
6 
6 

.477 
.478 
.479 
.480 
.481 
.482 
.483 

2999 
3006 

3013 
3020 

7 
7 
7 
7 
7 
7 
7 

9 

4 
5 
5 

1905 

2138 

2399 

2692 

1 

I9I0 

I9I4 
I9I9 

2143 
2148 

2153 

2404 
2410 
2415 

2698 
2704 
2710 

3027 
3034 
3041 

I 

7 

1 

.284 
.285 
.286 

1923 
1928 
1932 

5 
4 
4 

.334 
.335 
.336 

2158 
2163 
2168 

5 
5 

5 

.384 
.385 
.386 

2421 
2427 
2432 

6 
5 
6 

.434 
.435 
.436 

2716 

2723 
2729 

7 
6 
6 

.484 
.485 
.486 

3048 

3055 
3062 

7 

7 

7 

3 

4 
5 

6 

2 

3 
4 
4 

.287 
.288 
.289 

.290 

.291 
.292 
.293 

1936 
I94I 

1945 

5 
4 
5 
4 
5 
4 

.337 
•338 
.339 
.340 

•341 

.342 
.343 

2173 
2178 
2183 
2T88" 

5 
5 
5 
5 
5 
5 
5 

.387 
.388 

.389 

.390 

.391 
.392 
.393 

2438 

2443 
2449 

5 
6 
6 

5 
6 
6 

s 

.437 
.438 
.439 
.440 

.441 
.442 
.443 

2735 
2742 
2748 

7 
6 
6 

7 
6 
6 

7 

.487 
.488 

.489 

.490 

.491 
.492 
.493 

3069 
3076 
3083 

7 
7 
7 

7 
8 

7 

7 

7 
8 
9 

6 

1950 

^55_ 
2460 
2466 
2472 

27S± 
2761 
2767 
2773 

3090 

1 

1954 
1959 
1963 

2193 
2198 
2203 

3097 
3105 
3112 

I 
? 

8 

I 
2 

.294 
.295 
.296 

1968 
1972 
1977 

4 
5 
5 

.344 
.345 
.346 

2208 
2213 
2218 

5 

5 
5 

.394 
.395 
.396 

2477 
2483 
2489 

6 
6 
6 

.444 
.445 
.446 

2780 
2786 
2793 

6 

7 
6 

.494 
.495 
.496 

3119 

3126 

3133 

7 

3 
4 
5 
6 

2 

3 
4 
5 
6 
6 
7 

.297 
.298 
.299 

.300 

1982 
1986 
I99I 

4 
5 
4 

.347 
.348 
.349 
.350 

2223 
2228 
2234 

5 
6 

5 

.397 
.398 
.399 
.400 

2495 
2500 
2506 

5 
6 
6 

.447 
.448 
.449 
.450 

2799 
2805 
2812 

6 
7 
6 

.497 
.498 
•499 
.600 

3141 
3148 

3155 

7 
7 

7 

8 
9 

1995 

2239 

2512 

2818 

3162 

ANTIXOGARITHMS. 


log. 

no. 

d. 

log. 

no. 

d. 

log. 

no. 

d. 

log. 

no. 

d. 

log. 

no.  d. 

pp. 

.600 

.501 
.502 

.503 

3162 

8 
7 
7 
R 

.550 

.551 
•  552 
.553 

3548 
3556 
3565 
3573 

8 

9 

8 
8 

.600 
.601 
.602 
.603 

3981 

9 
9 
10 
q 

.650 

.652 
.653 

4467 

4477 
4487 
4498 

10 
10 
II 

TO 

.700 
.701 
.702 
.703 

5012  ^^ 

78 

3170 
3177 
3184 

3990 

3999 
4009 

S023  ^^ 
5035  [I 

5047 ;: 

III 
212 
322 

4  3  3 

.504 
.506 

3192 

3199 
3206 

7 
7 
8 

.554 
.555 
.556 

3581 
3589 
3597 

8 
8 
q 

.604 
.605 
.606 

4018 
4027 
4036 

9 
9 
10 

.654 

.655 
.656 

4508 
4519 
4529 

II 
10 
10 

.704 
.705 
.706 

5  44 

6  4  5 

7  5  6 

8  6  6 
967 

9 

.507 
.508 

.509 

.510 

•  5" 

.513 

3214 
3221 
3228 

3236 

3243 
3251 
3258 

7 
7 
8 

7 

8 

7 

8 

.557 
.558 
•559 
.560 

.562 
.563 

3606 

3614 
3622 

3631 

8 
8 
9 
8 

9 
8 

8 

.607 
.608 
.609 

.610 
.611 
.612 
.613 

4046 

4055 
4064 

4074 

9 
9 
10 

10 
9 

Q 

.657 
.658 

.659 

.660 

.661 
.662 
.663 

4539 
4550 
4560 

4571 

II 
10 
II 
10 

11 
II 

TO 

.707 
.708 
.709 
.710 

.711 
.712 
.713 

5093 ,, 
5105 11 
S117 

5129 

1  I 

2  2 

3639 
3648 
3656 

4083 

4093 
4102 

4581 
4592 
4603 

3  3 

4  4 

5  4 

6  5 

.514 
.516 

3266 

3273 
3281 

7 
8 
8 

•564 
.565 
.566 

3664 

3673 
3681 

9 

8 
f) 

.614 

.615 
.616 

41 1 1 
4121 
4130 

10 

9 

TO 

.664 
.665 
.666 

4613 
4624 

4634 

II 
10 
TT 

.714 
.715 
.716 

lilt  - 

5200 ;: 

7  6 

8  7 

9  8 

.518 

.519 

.520 

.521 
.522 

.523 

3289 
3296 
3304 

7 
8 

7 

8 

8 

7 
8 

.567 
.568 

.569 

.570 

•  571 

.572 
.573 

3690 
3698 
3707 

8 
9 

8 

9 
9 
8 
q 

.617 
.618 
.619 

.620 

.621 
.622 
.623 

4140 
4150 
4159 
4169 
4178 
4188 
4198 

10 

9 

10 

9 
10 
10 

q 

.667 
.668 
.669 

.670 
.671 
.672 
.673 

4645 
4656 
4667 

XI 
11 
10 
11 
II 
11 
TT 

.717 
•  718 
.719 
.720 
.721 
.722 
.723 

5212 

5224  " 

5236 " 

10 

1  I 

2  2 

3311 
3319 
3327 
3334 

3715 

4677 
4688 
4699 
4710 

5248 " 

3  3 

3724 
3733 
3741 

5260  " 
5272 " 
5284  ;^ 

\  I 

7     7 
3  8 

.524 
.526 

3342 
3350 
3357 

8 

7 

8 

.574 
.575 
.576 

3750 
3758 
3767 

8 

9 
n 

.624 
.625 
.626 

4207 

4217 
4227 

10 

TO 

0 

.674 
.675 
.676 

4721 
4732 
4742 

II 
TO 
TT 

.724 
.725 
.726 

5297  ,^ 
5309 11 

S32I  11 

3  9 
11 

.527 
.528 
.529 
.530 

.531 
.532 
.533 

3365 
3373 
3381 

8 

8 
7 
8 
8 
8 
8 

.577 
.578 
.579 
.580 

.582 
.583 

3776 
3784 
3793 

8 
9 
9 
9 
8 
9 

.627 
.628 
.629 

.630 

.631 
.632 

.633 

4236 
4246 
4256 

10 
10 
10 
10 
9 

TO 
TO 

.677 
.678 
.679 

.680 

.681 
.682 
.683 

4753 
4764 

4775 
4786 

4797 
4808 
4819 

II 
11 
11 
11 
11 
11 
T? 

.727 
.728 

.729 

.730 

.731 
.732 
•733 

5333 
5346  3 
5358  "  , 
S370  -  , 

5395  "  ' 

5408  ;3  <. 

r"  1 

2  2 

3  3 

^  4 

3388 

3802 

4266 
4276 
4285 
4295 

\     6 
3  7 
7     8 
^  9 
)  10 

3396 
3404 
3412 

3811 

3819 
3828 

.534 
.535 
.536 

3420 
3428 
3436 

8 
8 
7 

.584 
.585 
.586 

3837 
3846 

3855 

9 
9 

•  634 
.635 
.636 

4305 
4315 

4325 

10 
10 
TO 

.684 
.685 
.686 

4831 
4842 

4853 

II 
11 
TT 

.734 
.735 
•73^ 

5420 
5433 
5445  ;; ; 

12 

I 
I    2 

.537 
.538 
.539 
.540 

.541 

.542 
.543 

3443 
3451 
3459_ 
3467 
3475 
3483 
3491 

8 

8 
8 
8 
8 
8 
8 

.587 
.588 

.589 

.590 

.591 
.592 
•  593 

3864 

3873 
3882 

9 

9 

8 

9 
9 
9 
9 

10 
9 
9 

9 
9 
9 

.637 
.638 

.639 

.640 

.641 
.642 
.643 

4335 
4345 
4355 

10 
TO 
TO 
10 
10 
10 

.687 
.688 
.689 

.690 

.691 
.692 
.693 

4864 

4875 
4887 

4898 
4909 
4920 
4932 

11 
12 
II 
11 
11 
12 
TT 

.737 
.738 
.739 
.740 

.741 
.742 
.743 

5458  ,,  : 
5470  ,  j 
5483  "  I 

4 

5 

6 

>  7 

3890 

3899 
3908 

3917 

4365 

5495  "  ; 

5521 
5534  ]l 

8 

4375 
4385 
4395 

11 

13 

I 
3 
4 

.544 
.546 

3499 
3508 
3516 

9 
8 

8 

.594 
•595 
.596 

3926 
3936 
3945 

.644 
.645 
.646 

4406 
4416 
4426 

10 
10 
TO 

.694 

.695 
.696 

4943 
4955 
4966 

12 
11 
TT 

.744 
.745 
.746 

5546   , 
5559  \\  ^ 
5572   3 

.547 
.548 
.549 
.550 

3524 
3532 
3540 

8 
8 
8 

.597 
.598 
.599 
.600 

3954 
3963 
3972 

.647 
.648 

.649 

.650 

4436 
4446 

445_7_ 
4467 

10 
II 
10 

.697 
.698 
.699 

.700 

4977 
4989 
5000 

12 
11 
12 

.747 
.748 
.749 
.750 

5585   \ 
5598  \\   « 

5610  "  7 

5 
6 
8 

9 

3548 

3981 

5012 

5623  '^ 

12 

ANTirOGARITHMS. 


log. 

no. 

d 

log. 

no. 

d 

log. 

no. 

d.    log. 

no. 

d 

log. 

no. 

d.        pp. 

.750 

.751 
•  752 
.753 

5623 

13 
13 
13 

.800 
.801 
.802 
.803 

6310 

14 
15 
14 

.850 

.851 
.852 

.853 

7079 
7096 
7112 
7129 

.900 

17 

7943 

19 
18 
18 
19 

.950 

.951 

.952 
.953 

8913 

13(14 

^°    I     I     I 
21    2    3    3 
20     3     4     4 
Ji     4     5     6 

5636 

5649 
5662 

6324 
6339 
6353 

7962 
7980 
7998 

8933   , 
8954  . 
8974   . 

.754 
.755 
.756 

5675 
5689 
5702 

14 
13 
T3 

.804 
.805 
.806 

6368 
6383 
6397 

15 
14 

T5 

.854 
.855 
.856 

7145 
7161 

7178 

I   -906 

8017 
8035 
8054 

18 
19 
18 

.954 
•  955 
.956 

8995    . 
9016   ; 

9036 ; 

.1     6 

-I       n 

6     7 

8  8 

9  10 
0   II 

•757 
.758 
.759 
.760 
.761 
.762 
.763 

5715 
5728 

5741 

5754 

13 
13 
13 
14 
13 
13 
Trj 

.807 
.808 
.809 

.810 
.811 
.812 
.813 

6412 
6427 
6442 

15 
14 

.857 
.858 

.859 
.860 
.861 
.862 
.863 

7194 
7211 

7228 

r7     •9°7 

.909 
^9I3 

8072 
8091 
8110 

T9 

19 

18 

19 
19 
19 

TO 

.957 
.958 
.959 
.960 
.961 
.962 
.963 

9057 . 
9078 ; 
9099 ' 

I 

^        15  16 

I     I     2    2 

I     2    3    3 

3     4     5 

I     4     6     7 
1588 
I     6     9   10 

6457 

7244 

8128 

9120 

5768 
5781 
5794 

6471 
6486 
6501 

7261 

7278 
7295 

8147 
8166 
8185 

9141  , 
9162  ■ 

9183: 

.764 
.765 
.766 

5808 
5821 
5834 

13 
13 

.814 

.815 
.816 

6516 

6531 
6546 

15 
T5 

.864 
.865 
.866 

7311 

7328 

7345  , 

,  .914 
,  -915 
1 .916 

8204 
8222 
8241 

18 
19 
TO 

.964 
.965 
.966 

9226  ^^  9 ,4 ,4^ 
9247  ,, 

.767 
.768 

.769 

.770 

.771 

.772 
.773 

5848 
5861 

5875 

13 
14 
13 
14 
14 
13 

.817 
.818 
.819 
.820 

.821 
.822 
.823 

6561 
6577 
6592 

14 

15 
15 
15 
15 
16 

T5 

.867 
.868 
.869 

.870 

.871 
.872 
.873 

7362 

7379  , 
7396 

7413 

,  -917 
7 .918 

^  .919 

^   .920 

^    .922 
8    -9^3 

8260 
8279 
8299 

19 

20 

19 
19 
19 

19 
7.0 

.967 
.968 

.969 

.970 

.971 
.972 
.973 

9268 
9290  ^ 
9311 
9333 ' 

2       17  18 

J     I    2    2 
234 
2355 
14     7     7^ 
.     5     8     9* 
2     6   10   II 

^     7   12   13 

2       8     14     Id 

5888 

6607 

8318 

5902 
5916 
5929 

6622 
6637 
6653 

7430 
7447 
7464  ^ 

8337 
8356 
8375 

9354 1 
9376 
9397 : 

.774 
.775 
.776 

5943 
5957 
5970 

14 
13 

.824 
.825 
.826 

6668 
6683 
6699 

15 
16 

15 

.874 
.875 
.876 

7482 
7499 

7516 ; 

.924 

;  .925 

I    -926 

8395 
8414 

8433 

19 
19 

.974 
.975 
.976 

9419 

9441  ^ 
9462  ^ 

9   1 
2 

I 

7        11 

5!i6 
9  20 

.777 
.778 
.779 
.780 

.781 
.782 

.783 

5984 
5998 
6012 

14 
14 
14 
13 
14 
14 
Ti] 

.827 
.828 
.829 

.830 

.831 
.832 
.833 

6714 
6730 

6745 

16 

15 
16 

15 
16 
16 

.877 
.878 

.879 

.880 

.881 
.882 
.883 

7534 
7551 
7568  ' 

7586  1 
7603  ' 
7621  ^ 
7638  ; 

.927 
7    .928 

7  .929 

'   .930 

8  -931 
.932 

I    -933 

8453 
8472 
8492 

19 
20 

19 
20 
20 
19 

.977 
.978 
.979 
.980 
.981 
.982 
.983 

9484  ^ 
9506 1 
9528  ^ 

I 
2     2 
2      3 

^      51 
2       6    I 

2     7   I 

8   I 

2     9   I 

2  2 

4  4 
6     6 

8     8 

0  10 

1  12 

3  14 

5  16 

7I18 

6026 

6761 

8511 

9550 ' 
9572 1 
9594 1 
9616  ^ 

6039 

6053 
6067 

6776 
6792 
6808 

8531 
8551 
8570 

.784 
.785 
.786 

6081 
6095 
6109 

14 
14 

T5 

.834 
.835 
.836 

6823 
6839 
6855 

16 
16 

.884 
.885 
.886 

7656 

7674  ' 
7691  ' 

0    .934 
'    -935 
\    .936 

8590 
8610 
8630 

20 

20 

.984 
.985 
.986 

9638 
9661  ^ 
9683 1 

5        2 

2     I 
2     2 

1[22 

2    2 

.787 
.788 

.789 

.790 

.791 
.792 
.793 

6124 
6138 
6152 

14 
14 
14 
14 
14 
15 

.837 
.838 
.839 
.840 

.841 
.842 

.843 

6871 
6887 
6902 

16 
15 
16 
16 
16 
16 

t6 

.887 
.888 
.889 

.890 

.891 
.892 

.893 

7709 
7727  ^ 
7745  ' 
7762  ' 

«    -937 

.938 

^    .939 

B      ''' 

R    -941 

.942 

I    -943 

8650 
8670 
8690 
8710 

20 
20 
20 
20 
20 
20 

.987 
.988 

.989 

.990 

.991 
.992 
.993 

9705 
9727 1 
9750  ^- 
9772  ^ 

3 
^     4 
5     5   I 
2     ^   ^ 

7   1 
5     8   r 

z    9  ^ 

5     7 
B     9 

D    II 

3    13 

5    15 
7   18 
3   20 

6166 
6180 
6194 
6209 

6918 

6934 
6950 
6966 

7780  ' 
7798  ' 
7816  ' 

8730 
8750 
8770 

9795  ^' 
9817  : 

9840   ^; 

.794 
.795 
.796 

6223 
6237 
6252 

14 
15 

.844 
.845 
.846 

6982 
6998 
7015 

16 
17 
16 

.894 
.895 
.896 

7834 
7852  ' 
7870  ' 

,    -944 

•945 

]    .946 

8790 
8810 
8831 

20 
21 

.994 
.995 
.996 

9863 
9886   ^- 
9908   - 

.      I 
2 

'-      3 

23 
2 

5 
7 

.797 
.798 
.799 
.800 

6266 
6281 
6295 
6310 

15 
14 
15 

.847 
.848 

.849 

.850 

7031 

7047 
7063 

16 
16 
16 

.897 
.898 

.899 

.900 

7889 
7907  ^ 
7925  ' 

,    .947 
•948 
•949 
.950 

8851 
8872 
8892 

21 
20 
21 

•  997 
.998 
.999 
1000 

9931    ^. 

9954  ^' 
9977  '- 
0000  ^~ 

4 
5 

6 
9 

9 

12 
14 
16 

18 
21 

7079 

7943  ' 

8913 

536  UNIVERSITY    ALGEBRA. 

COMPUTATION  BY  I^OGARITHMS. 

695.  Cologarithms.  In  order  that  any  piece  of  work 
involving  multiplication,  division,  involution,  and  evolu- 
tion may  be  performed  by  the  addition  of  a  single  column 
of  logarithms,  the  Cologarithm,  instead  of  the  loga- 
rithm, of  a  divisor  is  written  down.  The  cologarithm, 
or  complementary  logarithm,  of  a  number  7i^  is  defined  to 
be  (10— log;^)-10. 

The  part  (10— log;e)  can  be  taken  from  the  table  just 
as  easily  as  log  n,  by  begin7iing  with  the  characteristic  and 
subtracting  in  order  all  the  figures  of  the  logarithm  from 
9,  except  the  last  figure  which  must  be  taken  from  10. 
Thus  log  256  is  given  in  table  as  2.4082,  whence 
•colog  256=7.5918-10.  It  is  plain  that  the  addition  of 
{10— log  ^^)  — 10  is  the  same  as  the  subtraction  of  log  n. 

The  convenience  arising  from  this  use  may  be  illus- 
trated as  follows  :  Suppose  it  is  required  to  find  x  from 
the  proportion  1193  :  ;r=749  :  1^^977     We  then  have 

log  1193  ==  3.0766 
^  log  .697  =  9.9216-10 
colog    749  =  7.1255-10 

\ozx       =0.1237 
Whence  .r=  1.329. 

EXAMPLES. 

Compute  the  values  of  the  following  expressions  by 
use  of  logarithms: 

I.     256x311x451.  2.     704  x  .21  x  .0649. 

7643x12.82 


3. 


864 


LOGARITHMS.  53/ 

Instead  of  adding  log  7643  and  log  12.82  and  subtracting  log  864 
from  the  sum,  the  work  should  be  done  as  follows: 
log  7643  =:  3.8833 
log  12.82  =  1.1079 
olog     864  =  7.0635-10 


2.0547 
antilog  2.0547  =  113.4. 


4.     61'^-r-17^  6.     158#^0.39.         8.     (^J|) 


515\4 


5.     ^4158.  7.     4in/613.  g.     ^(^-|)'- 

In  example  6,  log  0.39-9.5911-10.  Instead  of  ^  log  0.39 
=1(9.5911-10)  write  ^  log  0.39  =  1(49.5911 -50)=9.9182-10. 

10.^   (0.0641)o-o«4i^  11^     8.31-0-27. 

12.     (-0.412)~t. 

Solve  the  following  exponential  and  logarithmic  equa- 
tions : 

13.     5^=10.         14.     3^-2  =  5.  15.     53-2-^35^+4^ 

16.     23-+2^=5 

17.     log.,36=1.3678.  18.     Iog,2=logio4.933. 

19.  Find  the  amount  of  $550  in  15  years  at  5  per  cent, 
per  annum  compound  interest. 

20.'  What  should  be  paid  for  an  annuity  of  $100  a 
year  for  40  years,  money  being  supposed  to  be  worth  4^ 
per  cent? 

21.  A  corporation  is  to  repay  a  loan  of  $100,000  by  20 
equal  annual  payments.  How  much  will  have  to  be  paid 
each  year,  money  being  supposed  worth  5  per  cent  ? 


538  UNIVERSITY    ALGEBRA. 

22.  A  man  pays  $5  a  month  into  a  building  associa- 
tion which  nets  6  per  cent,  tor  8  years.  What  is  the 
value  of  his  stock  at  the  end  of  that  period  ? 

23.  The  population  of  the  United  States  in  1790  was 
3,930,000  and  in  1890  it  was  62,620,000.  What  was  the 
average. rate  per  cent,  increase  for  every  decade  of  this 
period  ? 

24.  Find  the  volume  of  a  cone  the  radius  of  the  base 
being  16.471  feet  and  the  altitude  8.644  feet. 

25.  Find  the  volume  and  surface  of  a  sphere  whose 
radius  is  11.927. 

26.  What  is  the  weight  in  tons  of  an  iron  sphere  whose 
radius  is  11.927  feet  if  the  weight  of  a  cubic  foot  of  water 
is  1000  ounces  and  the  specific  gravity  of  iron  is  7.21  ? 

KXPONKNTIAL  AND  LOGARITHMIC  S:e:rIKS. 

696.     The  expression  (IH — j     may  be  expanded  by 

the  binomial  theorem  no  matter  what  value  x  may  have, 

provided  only  that  —  is  less  than  unity;  that  is,  provided 

n^\.     Such  expansion  gives 

1     7ix{nx—V)  1      7ix{rix -X){71X —^  1 

'^^n^      172     '7^'^  TT.-g        ^"^'^ 

,  nx{nx—X)'  '  '{jix—r-^X)  1  . 

+  Yr_  ;  n^^  '  '  '       ^^^ 

which  may  be  put  in  the  form 

(1+-)  =n-^+_^-2-+ 


(■+s"-- 


1.2.3 


<^-J)- -(^-V) 


\r 


LOGARITHMS.  539 

This  being  true  for  all  values  of  x,  we  may  put  ;r=  1 . 
This  gives 

v^n)  -^+^+t:2  + — rxs — +  •  ■  • 

■^    ""'    ^    jlJ+...     (2). 


Hence,  from  (1)  and  (2) 

xix )  xix ).  .  Ax I 

=  1+^+      -^2     +  •  •  •  "^ \? +•••  (^> 

Since  this  equation  is  true  for  all  values  of  n  greater 
than  unity,  it  is  true  if  n  increases  in  value  without  limit. 

1       2 
Whence,  since  the  limit  of  —  or  — >  etc,  is 0  as  ;2  increases 

71       n 

without  limit,  we   have,  by  taking  the  limits  of  both 
members  of  (3)  as  n  increases,  (Art.  420), 
/  11  1  \'' 

(i+i+[2  +  L3  +  ---+jy+---)- 

The  convergent  series  in  the  left  member  of  (4)  is 
usually  represented  by  e,  and  is  called  the  Natural  or 
Naperian  Base. 

Thus  we  have,  for  all  values  of  x, 

This  is  the  Exponential  Series  for  the  base  e. 


S40 


UNIVERSITY    ALGEBRA. 


In  equation  (3)  above  it  is  plain  that  the  numerator  in  the  rth  term 
M Yl J  .  .  .M ~—]  approaches  1  as  a  limit  if  r  is  finite. 

As  r  increases  without  limit  we  do  not  know  from  anything  yet  con- 
sidered that  this  infinite  product  approaches  1.  But  this  is  a  matter 
that  need  not  concern  us,  for  as  the  series  is  convergent  for  all  the  values 
of  //>1,  its  terms  are  known  to  ultimately  decrease  in  value  without 
limit.  So  we  shall  leave  the  question  open  whether  the  numerators 
are  ultimately  1  or  not,  knowing  that  the  terms  ultimately  decrease 
without  limit. 

The  value  of  e  may  be  readily  computed  as  follows  : 


2 

1.000000 
1.000000 

3 

0.500000 

4 

0.166667 

5 

0.041667 

6 

0.008333 

7 

0.001389 

8 

0.000198 

9 

0.000025 

0.000003 


Adding  ^=2.718282 

This  happens  to  be  correct  to  6  places,  as  calculation  using  more 
places  would  show,  but  an  error  in  the  last  place  or  two  may  be 
usually  expected  owing  to  the  neglected  figures. 

697.  Any  Base.  In  equation  [11]  above  substitute 
ex  for  X.     We  then  have 

Let  a=e^,  so  that  also  c=\og,a.  We  then  have,  by- 
substitution, 

x\\ogAy 


.^=l4-;.log.a+?-'(^"^^"^' 


•  •+- 


[12] 


|2  r 

which  is  often  called  the  Exponential  Series  or  Theo- 
rem. 


LOGARITHMS.  54I 

698.  The  Logarithmic  Series  is  the  expansion  of 
logX^+x)  in  terms  of  the  ascending  power  of  ;r. 

Take  the  exponential  series 

a^=l+ylog.a+^-^^^+^-^'^+    ..         (1) 

Whence,  transposing  the  1  and  dividing  through  byjc, 
_-=log.a+^-^-  + -—+  .  .  .  j        (2) 

Therefore,  since  these  variables  are  always  equal, 
Whence  it  is  easy  to  see 

Now  put  l+;t:  for  a.     Then  we  have 

limit/O+fVjzi^ 

Expanding  (l+xY  by  binomial  formula, 

+^^=^^^-=^''+ ■ )  (=) 

in  which  x  must  not  be  greater  than  1,  since  the  binomial 
formula  has  been  used. 

The  limit  of  the  right  member  as  j^/^  0  can  be  plainly 
seen;  whence  we  obtain  the  equation 

logXl+^)=^-^2-+  3"     4-+  •  •  •  +  — ~^  +  -  ■  -[13] 
This  is  the  Logarithmic  Series. 

699.  Convergency  of  the  Series.  The  above  series 
is  not  convergent  for  values  of  x  greater  than  1 ,  and  hence 
cannot  be  used  for  computing  the  logarithm  of  any  integral 


,       ..^   ,     limit/(l+^)--l\ 


\ozl\-x)=-x — -- — o— — J- (2) 


542  UNIVERSITY    ALGEBRA. 

number  but  2.  The  following  scheme  will  give  a  series 
which  is  available  for  computing  the  logarithms  of  all 
integers. 

700.  Logarithmic  Series  Convergent  for  Integral 

Values  of  x.     In  the  logarithmic  series 

logXl  +  x)=x-^2'  +  ~3~-T+  ■  ■  ■  (^) 

substitute  —x  for  x  and  we  shall  have 

"2"      3"~~4 
Subtracting  (2)  from  (1),  observing  that 

\-\-x 
log/l +  -^0~^ogXl~--^)=lo§'^T3~'    we  obtain 

1    ,     ,  ^   2^+2 .      2^ 

Now  put  x—-^ — —T)  whence  !+;»;=  5 — -—,\-x=r: — — r> 

and:; = Therefore  we  obtain 

\—x       z 

l+^_^/     1  1  1  \ 

^^^   ^        V2^+l'^3(2^+l/"^5(2^+iy"^'  *  J     ^^ 

Whence,  since  log^ =log/l4-'2')— log^-s'.  by  substitut- 

z 

ing  and  transposing  log^-a-  we  have 

logXl+^)=log^+2(^j  +  3^J^^^3+g^J^^^,  +  ..)(5) 

This  series  converges  rapidly  for  integral  values  of  z. 
Its  use  in  computing  the  logarithms  of  numbers  will  now 
b)e  explained. 

701.  To  Compute  the  Natural  Logarithms  of 
Numbers.  The  logarithm  of  1  is  0  in  all  systems.  To 
compute  log^2,  put  z=  1  in  equation  (5)  above.  We  then 
^obtain 


LOGARITHMS.  543 

l°^'2=2(l+3-L  +  5-l,+^4-,+,4,+  .  .  .)=.6931472 

Now  put  ^=2  in  equation  (5).     Then  we  have 

log.3=.6931472+2(^+3^+g^  +  ^+.  .  .)=1.0986123 

To  find  log,4  we  know  log^4=log,22  =  2  log^2;  whence 

log,4=  1.3862944 
To  find  log^5,  put  ^=i  in  equation  (5).   We  then  have 

log,6=1.3862944+(i+gi^3  +  g^+^+  ■  ■  .)=1.6094376 

In  like  manner  the  logarithms  of  all  numbers  may  be 
found.  The  logarithms  of  composite  numbers  need  not 
be  computed  by  the  series,  since  the  logarithm  of  any 
composite  number  can  be  found  by  adding  the  logarithms 
of  its  component  factors. 

702.  Logarithms  of  a  Number  in  Different  Sys- 
tems. Consider  the  systems  whose  bases  are  a  and  e. 
Then,  if  n  is  any  positive  number,  we  wish  to  find  the 
relation  between  log^^^  and  logji. 

Let  x=logen  and  y=\ogji. 

Then  7t=e^  and  n^=^a^\ 

whence,  e''=^a^.  (1) 

X 

Therefore,  a^e~y.  (2) 

If  we  write  this  in  logarithmic  notation,  we  have 

log,^=y  ,  (3) 

or,  substituting  the  values  of  x  and  y,  we  obtain 

log^;^  ^  ^ 

Therefore  loga^=:j log,;^,  (5) 

703.  Modulus    of   Common    Logarithms.     If,  in 

equation  (5)  above,  we  understand  e  to  represent  the 


544  UNIVERSITY   ALGEBRA. 

Natural  base  and  a  the  common  base,  then  equation  (5) 

becomes  log  n=- rr^log^^^.  (1) 

But  log40=log,2+log,5=(by  Art.  701)  2.3025851  and 

— j^=. 43429448.     Therefore,  representing    .43429448 

by  M,  we  have 

log  n=Mlog,n,  (2) 

The  decimal  represented  by  M  is  known  to  282  decimal 
places  and  is  called  the  Modulus  of  the  system  of  com- 
mon logarithms. 

Equation  (2)  is  seen  to  express  the  important  truth 
that  ^ke  common  logarithm,  of  any  ftumber  can  be  obtainei 
by  multiplying  the  natural  logarithm  of  that  number  by  thL 
modulus  of  the  common  system. 

704.     Computation  of  Common  Logarithms.  We 

can  now  compute  the  common  logarithms  of  numbers. 
We  merely  need  to  multiply  each  of  the  Natural  loga- 
rithms already  found  by  the  modulus  .43429448-  •  .  In 
this  manner  we  find 

log  2=0.3010300 

log  3=0.4771213 

log  4=0.6020600 

log  5=0.6989700 
etc.  etc. 

How  can  you  find  log  6  ? 

Historical  Note. — The  almost  miraculous  power  of  modem 
calculation  is  due  to  three  inventions  :  the  Arabic  Notation,  Decimal 
Fractions,  and  Logarithms.  The  invention  of  logarithms  in  the  first 
quarter  of  the  17th  century  was  admirably  timed,  for  Kepler  was 
then  studying  planetary  orbits  and  Galileo  had  just  turned  his  telescope 
toward  the  stars.  It  has  been  said  that  the  invention  of  logarithms, 
"by  shortening  the  labors,  doubled  the  life  of  the  astronomer."  The 
honor  of  invention  belongs  to  a  Scotchman,  John  Napier,  Baron  of 
Merchiston.       His   first   work,    the   Mirifici    logarithmoru?n   canonis 


LOGARITHMS.  545 

descriptio,  1614,  contains  a  table  of  natural  sines  and  their  logarithms 
to  seven  decimal  places.  Napier's  logarithms  were  not  the  same  as 
the  Natural  Logarithms.  The  base  of  his  logarithmic  number  was 
not  e,  but  nearly  ^-i.  The  base  required  by  his  reasoning  is  exactly 
^~i.  It  must  be  remembered,  however,  that  Napier  never  connected 
logarithms  with  the  idea  of  a  base.  This  concept  was  introduced 
later.  To  us  who  know  how  naturally  logarithms  flow  from  the 
exponential  symbol,  it  seems  curious,  indeed,  that  logarithms  should 
have  been  invented  before  exponents  had  come  into  general  use. 
Retnemher  that  integral  exponents  (as  we  now  have  them)  ivere  first 
used  by  Decartes,  1637,  while  negative  and fractio7ial  exponents  luere first 
used  by  Joh?i  Wallis  in  1656,  and  literal  exponents  by  Nezvton  in  1676. 

Napier's* conception  of  logarithms  was  quite  different  from  ours, 
and  is  contained  in  the  meaning  of  the  term  itself,  which  comes  from 
two  Greek  words  meaning  themunber  of  the  ratios.  This  idea  of  a  loga- 
tithm  may  be  thus  explained  :  Suppose  the  ratio  of  1  to  10  be  divided 
into  a  large  number  of  equal  ratios  (or  factors),  say  1,000,000.  Then 
it  is  true  that  the  ratio  of  1  to  2  is  composed  of  801030  of  these  equal 
ratios,  and  301030,  the  nwjiber  of  the  ratios,  is  the  logarithm  of  2.  In 
the  same  way,  the  ratio  of  1  to  3  is  composed  of  477121  of  these  equal 
ratios,  and  the  logarithm  of  3  is  hence  said  to  be  477121. 

His  method  of  computing  them,  though  ingenious,  was  tedious.  In 
fact,  the  great  work  of  computing  logarithmic  tables  was  completed 
before  the  discovery  of  the  logarithmic  series.  Napier  made  log 
10''=0  and  caused  his  logarithmic  figures  to  increase  as  the  numbers 
themselves  decreased.  This  is  shown  by  the  following  arithmetic  and : 
geometric  progressions: 
Napier' s  logaritlwis,  0,  1,  2,  ...  n. 


Numbers 


m,  lo^(i-iif),  lo'(i-i^)  •  ■  •  io'(i-i[.)"- 

The  relation  between  Napier's  and  natural  logarithms  is  expressed 
by  the  following  formula: 

10^ 
log;v^.y=10^  \og  — 

The  news  of  the  wonderful  invention  of  logarithms  induced  Henry 
Briggs,  professor  of  Gresham  College,  London,  to  visit  Napier  in  1615. 
Briggs  suggested  to  him  the  advantages  of  a  system  in  which  the 
logarithm  of  1  should  be  0  and  the  logarithm  of  10  should  be  1. 
Napier,  having  already  thought  of  this  change,  encouraged  Briggs  to. 

35  —  U.  A. 


546  UNIVERSITY    ALGEBRA. 

compute  a  system  of  new  logarithms  and  made  many  important  sug- 
gestions, among  which  was  that  of  keeping  the  mantissas  of  all  loga- 
rithms positive,  by  using  negative  characteristics.  In  1G17  Briggs 
published  the  common  logarithms  of  the  first  1000  numbers,  and  in 
1624  the  common  logarithm  of  numbers  from  1  to  20000,  and  from 
90000  to  100000,  to  14  decimal  places.  The  gap  between  20000  and 
90000  was  filled  up  by  Adrain  Vlacq,  of  Gouda  in  Holland,  who  pub- 
lished 1628  the  logarithms  of  numbers  from  1  to  100000  to  ten  places. 
Vlacq' s  table  is  the  source  from  which  nearly  all  the  tables  have  been 
derived  which  have  been  published  since. 

The  first  calculation  of  logarithms  to  the  base  of  the  natural  sys- 
tem was  made  by  John  Speidell  in  his  New  Logarithms,  published  in 
London  in  1616. 

Problem. — Prove  that  the  base  of  Napier's  logarithmic  numbers  is 
nearly  ^-  l ,  by  taking  (in  the  arithmetical  and  geometric  progressions 
given  above),  ;?— lO'',  dividing  each  term  in  both  progressions  by  10^ 
and  calculating  the  value  of  the  term  in  the  new  geometric  progression, 
whose  logarithm  is  unity.-  Then  deduce  the  formula,  given  above, 
expressing  the  relation  between  Napier's  and  natural  logarithms. 


CHAPTER  XXXI. 

COMPI.KX  NUMBERS. 

705.  Arithmetical  and  Algebraic  Numbers.  Let 
us  notice  what  is  meant  by  Algebraic  Number,  and  how 
the  notion  of  it  may  originate.  The  primitive  conception 
of  number  is  used  when  we  enumerate  the  marbles  in  a 
box,  and  say:  0,  1,  2,  3,  4,  etc.  Our  simple  scale  is 
atithnietical  number^  and  it  runs  down  to  a  definite 
nothing,  or  zero,  and  stops.  But  let  us  attempt  to  apply 
this  scale  in  the  measurement  of  other  things.  Suppose 
we  are  estimating  time,  where  is  the  zero  from  which 
all  time  is  to  be  measured  in  one  ** direction"  or  sense? 
There  is  no  such  zero,  as  in  the  case  of  marbles ;  for  we 
can  conceive  of  no  event  so  far  past  that  no  other 
events  precede  it.  We  are  forced  to  select  a  standard 
event,  and  measure  the  time  of  other  events  with  refer- 
ence to  the  lapse  before  or  after  that.  The  zero  used  is 
an  arbitrary  one,  and  there  is  quantity  in  reference  to  it 
in  two  opposite  senses,  future  and  past,  or,  as  is  said  in 
algebra,  positive  and  7iegative.  We  are  likewise  obliged 
to  recognize  quantity  as  extending  in  two  opposite  senses 
from  zero  in  the  attempt  to  measure  many  other j things; 
in  locating  points  along  an  east  and  west  line,  no  point  is 
so  far  west  that  there  are  no  other  points  west  of  it,  hence 
could  not  be  located  in  the  arithmetical  scale ;  the  same 
in  measuring  force,  which  may  be  attractive  or  reptdsive; 
or  motion,  which  may  be  tozvard  or  fro ?n,  etc.  Thus  our 
notion  of  algebraic  quantity,  as  we  name  this  kind  of 
quantity,  has  arisen. 


548  UNIVERSITY   ALGEBRA. 

Because  of  the  peculiar  analogy  between  our  notion  of 
time  and  algebraic  quantity,  algebra  has  been  called  the 
science  of  pure  time.  All  quantities  are  measured  exactly 
as  a  past  and  future,  or,  graphically,  along  a  line,  in  both 
directions  from  a  zero  point.  Fixing  our  attention  on 
any  event,  time  exists  in  one  sense  (future)  and  in  exactly 
the  opposite  sense  ^past)  and  in  no  other  sense  at  all. 
Likewise  with  algebraic  numbers,  we  never  get  <9z^of  the 
line.  This  kind  of  quantity,  although  more  general  than 
arithmetical  number,  is  really  quite  restricted.  We  ob- 
serve, at  once,  that  there  is  an  opportunity  of  enlarging 
our  conception  of  quantity  if  we  can  only  get  out  of  our 
line,  or  ''one-way  spread''  as  some  say,  and  explore  the 
region  without.  We  may  seek,  then,  an  extension  of  our 
notion  of  quantity  which  will  enabJe  us  to  consider,  along 
with  the  points  of  our  line,  those  which  lie  without. 

706.  Numbers  as  Operators.  We  usually  distin- 
guish in  algebra  between  symbols  of  number  or  quantity 
and  sy7nbols  of  operation.  Thus  a  symbol  which  may  be 
considered  as  answering  the  question  "how  many?"  or 
*'how  much?"  is  called  a  number  or  quantity,  while  a 
symbol  which  tells  us  to  do  something  and  which  may  be 
read  as  a  verb  in  the  imperative  mood,  is  called  a  symbol  of 
operation  or  simply  an  Operator.  Thus,  in  V  2,  when 
read  "take  the  square  root  of  two' ' ,  we  distinguish  readily 
the  S3^mbol  of  operation  from  the  symbol  of  number. 
Likewise  in  log  21  we  may  look  upon  "log,"  the  symbol 
for  "find  the  logarithm  of",  as  a  symbol  of  operation 
and  21  as  a  symbol  of  number  or  quantity. 

It  is  interesting  to  note  that  any  number  may  be  re- 
garded as  a  symbol  of  operation;  and  that  thereby  some 
original  conceptions  may  be  very  conveniently  extended. 
Thus,   10   may  be  regarded  not  only  as  ten,  answering 


COMPLEX    NUMBERS.  549 

the  question  ''how  many?",  but  as  denoting  the  ^/(?r^- 
tion  of  taking  unity,  or  whatever  follows  it,  ten  times;  to 
express  this,  we  may  write  10  1,  in  which  10  may  be 
called  a  teyisor,  (that  is,  ''stretcher''^  or  a  symbol  of  the 
operation  of  stretchiyig  a  unit  until  the  result  obtained  is 
ten  fold  the  size  of  the  unit  itself.  In  the  same  way,  the 
symbol  2  may  be  looked  upon  as  denoting  the  operation  of 
doubling  unity,  or  whatever  follows  it;  likewise,  the  tensor 
3  may  be  looked  upon  as  a  trebler,  4  as  a  qiiadrupler^  etc. 

With  the  usual  understanding  that  any  symbol  of 
operation  operates  upon  what  follows  it,  we  may  have  com- 
pound operators  like  2.2.3.  Here  3  denotes  that  unity 
is  to  be  trebled,  2  denotes  that  this  result  is  to  be  doubled, 
and  2  denotes  that  this  result  is  to  be  doubled.  Thus 
representing  the  unit  by  a  line  running  to  the  right,  we 
have  the  following  representation  of  the  operators  : 

The  unit  — > 

3.1  —>  —  >—> 

2.3.1  > > 

2.2.3.1  > > 

Notice  now  the  extended  significance  of  an  exponent. 
It  means  to  repeat  the  operation  designated  by  the  base; 
that  is,  the  operation  designated  by  the  base  is  to  be 
performed,  and  performed  again  on  the  result,  and  per- 
formed again  on  this  result,  and  so  on,  the  number  of  oper- 
ations being  denoted  by  the  exponent.  Thus  10  ^  means 
to  peform  the  operation  of  repeating  unity  ten  times 
(indicated  by  10)  and  then  to  perform  the  operation  of 
repeating  the  result  ten  times,  that  is,  10(10-1).  Also, 
10^  means  10[10(10  •  1)].  Then,  of  course,  the  exponent 
zero  can  only  mean  that  the  operation  on  unity  denoted 
by  the  number  is  not  to  be  performed  at  all;  that  is,  unity 
is  to  be  left  unchanged;  thus  10^  or  10^  •  1  =  1. 

The  expression  V  4,  looked  upon  as  a  symbol  of  oper- 
ation, denotes  an   operation  which  must  be  performed 


550  UNIVERSITY   ALGEBRA. 

twice  to  qnadrnple;  that  is,  such  that  (1^4)  2  =  4.  lyikewise, 
f'  4  denotes  an  operation  which  must  be  performed  three 
times  in  succession  in  order  to  be  equivalent  to  quad- 
rupling. 

We  know  that  the  operation  denoted  by  2,  if  performed 
twice,  is  equivalent  to  quadrupling,  therefore  1^4=2,  etc. 
Just  as  42,  4^,  etc.^may  be  called  stro7ige7  tensors  than 
a  single  4,  so  l/4,  1^4  may  be  called  weake?  tensors 
than  the  operator  4. 

The  expression  —1,  looked  upon  as  a  symbol  of  oper- 
ation, is  not  a  tensor,  as  it  leaves  the  size  unchanged  of 
what  it  operates  upon.  But  if  applied  to  any  quantity  it 
will  change  the  sense  in  which  the  quantity  is  then  taken 
to  exactly  the  opposite  sense.  Thus,  if  -f  6  stands  for 
six  hours  after,  then  (— l)(-f  6)  stands  for  six  hours  before 
a  certain  event,  and  —1  is  the  symbol  of  this  operation 
of  reversing.  Also  if  (4-6)  stands  for  a  line  running  six 
units  to  the  right  of  a  certain  point,  then  ( — 1)(  +  6) 
stands  for  a  line  running  six  units  to  the  left  of  that 
point,  so  that  (—1)  is  the  symbol  which  denotes  the 
operation  of  turning  a  straight  line  through  180°.  As 
2,  3,  4,  etc.,  were  called  tensors  when  looked  upon  as 
symbols  of  operation,  we  may  conveniently  designate  the 
operator  —1  as  the  reversor, 

707.  Imaginaries.  We  remember  that  such  expres- 
sions as  3-f  1/— 5,  c+V^—d,  j/  — 24,  etc.,  were  forced 
upon  our  notice  in  the  solution  of  quadratic  equations. 
It  is  customary  to  call  such  Imaginaries,  because  of  the 
presence  in  them  of  a  term  like  V  —a,  which,  evidently, 
does  not  correspond  to  any  algebraic  number  whatever. 
But,  remembering  the  restricted  nature  of  algebraic  num- 
ber, it  is  possible  that  such  expressions  are  unreal  only 
in  an  algebraic  sense;  that  it  the  restriction  can  be  re- 


COMPLEX    NUMBERS.  551 

moved  by  an  extension  of  our  conception  beyond  a  mere 
linear  or  past  and  future  notion  of  quantity,  the  expres- 
sion may,  perhaps,  become  as  ''real"  as  algebraic  numbers 
now  are.  

Although  V  —  1  cannot  consistently  with  the  meanings 
of  V  and  —1  be  looked  upon  as  answering  the  question 
''how  many?"  or  "how  much?"  and  therefore  is  not  an 
algebraic  number,  yet  if  we  consider  it  as  a  symbol  of 
ope7'ation,  it  can  be  given  a  meaning  consistent  with  the 
operators  already  considered.  For  if  2  is  the  operator 
that  doubles  and  t/2  is  the  operator  that  when  used 
twice,  doubles,  then  if  —1  is  the  operator  that  reverses, 
the  expression  V  —  1  should  be  an  operator  that  when 
used  twice,  reverses.  So,  as  —1  may  be  defined 
as  the  .symbol  which  operates  to  turn  a  straight  line 
through  an  angle  of  180°,  in  a  similar  way  we  define  the 
expression  v  — 1  ^^  that  symbol  which  denotes  the  operation 
of  turning  a  straight  line  through  aji  angle  of  90°  in  the 
positive  directio7i. 

It  is  customary  in  mathematics  to  consider  rotation 
opposite  to  that  of  the  hands  of  a  watch  as  positive  rota- 
tion. The  restriction  of  positive  rotation  is  inserted  in 
the  definition  merely  for  the  sake  of  convenience. 

708.  Graphic  Representation,  In  figure  16,  let 
a  be  any  line.  Then  a  operated  on  by  l/  — 1,  that  is, 
V  —  \'a  is  a  turned  tip,  or  positively,  through  90°,  which 
gives  OB,  Now,  of  course,  V —\  can  operate  on  V  —\'a 
just   as   well  as  on  a.     Then  l/— l(l/— 1  •  a)  or  OC  is 

is  V —\   a    or    OB     turned positively     through     90°. 

y  ^[l/^(y  —  1  •  a)]  is  l/— l(l/—l .  a)  turned  through 
90°,  etc. 

As  we  are  at  liberty  to  consider  two  turns  of  90°  the 
same   as   one  turn  of  180°,.*.   V —\{V —\  •  d)=(^—X)a. 


552 


UNIVERSITY   ALGEBRA. 


Also  OD=i-V)OB,  .■.On=-0/-l.a),  but>^-l(-a) 
=  0B,  .-.  -(i/-l.«)  =  i/:^(_a).  Thus  the  student 
may  show  many  like  relations. 

B 


I  I 

_      \ 


\ 


<8 


I 


Fig.  i6. 

The  operator  v—1  is  usually  represented  by  the  sym- 
bol /  and  will  generally  be  so  represented  in  what  follows: 

KXAMPI^KS. 

Interpret  each  of  the  following  expressions  as  a  symbol 
of  operation : 

1.  2,  3,  4,  -1. 

2.  32,    23,   40,    (-1)2,    (-1)5. 

3.  i/2,  1/3,  V^,  i/2,  r^. 

Select  a  convenient  unit  and  construct  each  of  the  fol- 
lowing expressions  geometrically,  explaining  the  meaning 
of  each  operator: 

4.  2-3-51.  7.     (-1)2. 1/^1. 

5.  23(-l)-l.  8.     22(-l)3(i/Zi)o.i, 

6.  3t/^-2  1.         9.     Sl/^(-l)l/^l. 


'  COMPLEX    NUMBERS.  553 

709.  Laws.  It  is  implied  in  the  definition  above 
that  the  operator  i  must  follow  the  ordinar}^  laws  of 
algebra  as  set  forth  in  Arts.  107-113.  The  requirement 
of  these  laws  completes  the  definition  of  i  given  above. 
The  following  are  illustrations  of  each  law: 

Commutative  Law  : 

c-\-  di-\-  a  +  bi=  c-\-  a  +  di-\-  bi=  di-\-  c+  bi+  a,  etc. 
az=ia^  iai=ua=au^  etc. 

The  equation  lOj/ -l-j/-!  •  10,  or  better  lO]/^-!-! 
=  |/'  —  1  .  10  •  1  may  be  said  to  mean  that  the  result  of  perfonnUtg  the 
operation  of  turning  unity  through  OO'^  and performiiig  upon  the  result 
the  operation  of  takijzg  it  ten  times  is  the  same  as  the  result  of  perform- 
ing the  operation  of  takittg  tinity  ten  tifues  and  performing  upon  this 
result  the  operation  of  turnitts^  through  90®. 

Associative  Law : 

(c-\-di')-{-{a-\-bi^  =  c+(di+d)  +  bi,  etc. 
(ab')i==a(bi)  =  abi,  etc. 
Distributive  Law: 

(a-\-b)i=ai+bi,  etc. 

710.  The  special  relation     

V~^''  =  a\/-1,  [1] 

which  follows  necessarily  from  the  above,  is  very  im- 
portant.    In  other  symbols  it  may  be  written 

in  which  form  it  is  seen  to  be  an  application  of  one  of  the 
index  laws.  It  may  be  deduced  from  the  laws  above,  as 
follows. 

(z^a")  2"=  (iiad)'^=^  (m/a)  2"=  \_{id)  (z^)]  2"=  ia. 

By  means  of  this  relation,  we  put  expressions  like 
V"'— 3,  1/— 4,  V—b,  etc.,  in  the  forms  zl/3,  22,  z|/^ 
etc.  In  what  follows  it  is  presupposed  that  all  such  ex- 
pressions are  reduced  to  this  form. 

The  relation  "l/-4  =2  j/  — 1  may  be  interpreted  as  follows:  (—4) 
is  the  operator  that  quadruples  and  reverses;  then  j,/  — 4  is  an  opera- 


554  UNIVERSITY   ALGEBRA. 

tor  which  used  twice  quadruples  and  reverses.     But   2v^  — 1  is  an 
operator  such  that  two  such  operators  quadruple  and  reverse.  That  is, 

711.  Typical  Form.  It  will  be  shown  in  the  follow- 
ing theorems  that  any  expression  containing  both  real 
numbers  and  imaginaries  may  be  put  in  the  form  a  +  bi, 
in  which  both  a  and  b  are  real.  The  expression  a  +  bi  is 
therefore  said  to  be  the  Typical  Form  of  the  imaginary. 
An  expression  of  the  form  a-{-bi  is  also  called  a  Com- 
plex Number,  since  it  contains  a  term  taken  from  each 
of  the  following  scales,  so  that  the  unit  is  not  single  but 
double  or  complex : 

._3,  -2.  -1,0,  +1,  +2,  +3,... 
-  .  — 3^,  —2/,  — /,  0,  +^;  +2/,  +3/, .  .  . 
It  is  important  to  note  that  the  only  element  common 
to  the  two  series  in  this  complex  scale  is  0. 

712.  Graph  of  a  Complex  Number.  Any  real 
number,  or  any  expression  containing  nothing  but  real 
numbers,  may  be  considered  as  locating  a  point  in  a  line. 

Thus,  suppose  we  wish  to  draw  the  expression  2  +  5. 
Let  O  be  the  zero  point  and  OX  the  positive  direction. 
Lay  off  0A  =  2  in  the  direction  OX  and  at  A  lay  off 
AB=5  in  the  direction  OX.  Then  the  path  OA  +  AB  is 
the  geometrical  representation  of  2-f  5. 

O A B  X 

Any  complex  number  may  be  taken  as  the  representa- 
tion of  the  position  of  a  point  in  a  plane.  For,  suppose 
c-\-di\s  the  complex  number.  Let  O  be  the  zero  point 
and  OX  the  positive  direction.  Lay  off  OA=-\-c  in 
the  direction  OX  and  at  A  erect  di  in  the  direction  OY, 
instead  of  in  the  direction  OX  as  in  last  example.  Then 
c-\-di  defines  the  position  of  the  point  P  with  reference  to 


COMPLEX   NUMBERS. 


S5S 


(9,  and  the  path  OA+AP,  or  OP,  is  a  geometrical  repre- 
sentation of  c-\-di.     In  the  same  manner  C'—diy   —c—di 

and  —c+di  may  be  constructed. 
Y 


Fig.  17. 

713.  The  meaning  of  some  of  the  laws  of  algebra  as 
applied  to  imaginaries  may  now  be  illustrated.  I,et  us 
construct  c-^-di-^-a-i-bi, 

a  c 


Y 

F\ 

G 

P 

'^ 

+ 

I             h\ 

B 

a 

c 

b 

^ 

^ 

0 

E\ 

d 

D 

X 

J 

C 

A 

F 

IG. 

18. 

The  first  two  terms,  c-\-di,  give  OA  +  AB,  locating  B. 
The  next  two  terms,  a  +  di,  give  BC+CP,  locating  P. 


5S6  UNIVERSITY    ALGEBRA. 

Hence  the  entire  expression  locates  the  point  P  with 
reference  to  O.  Now  if  the  original  expression  be  changed 
in  any  manner  allowed  by  the  laws  of  algebra,  the  result 
is  merely  a  different  path  to  the  same  point.     Thus: 

c+a  +  di-\-bt  is  the  path  OA,  AD,  DC,  CP, 
(^+^)+  {d+b)i  is  the  path  OD,  DP. 

a  +  dl-\-c^-biis>  the  path  OE,  EH,  HC,  CP. 

a  +  dz-{-bi+cis  the  path  OE,  EH,  HE,  EP,  etc. 

The  student  should  consider  other  cases.  Are  there 
any  methods  of  locating  P  with  the  same  four  elements, 
which  the  figure  does  not  illustrate? 

714.  Powers  of  i.  We  shall  now  interpret  the  powers 
of  z  by  means  of  the  new  significance  of  an  exponent  and 
by  the  commutative,  associative  and  other  laws.     First: 


i^  or  /o  •  1 

=  +  1. 

i^  or  i^  '  1 

=      /. 

z2 

=  —  1. 

i^  =  pt 

=  — /. 

i^=.iH'' 

=  +1. 

i^  =  iH 

=      i. 

i^  =  i^z 

=  — 1. 

V  =  iH 

=  —  /. 

i^^Vi 

=  +  1. 

etc.  etc. 

Whence  it  is  seen  that  all  even  powers  of  /  are  either 
+  1  or  —1,  and  all  odd  powers  are  either  /  or  —i.  The 
student  may  reconcile  this  with  figure  16. 

715.  Two  complex  numbers  are  said  to  be  Conjugate 
if  they  differ  only  in  the  sign  of  the  term  containing  V  —  1. 
Such  are  x+ry  and  x—ry. 


COMPLEX    NUMBERS.  557 

716.  Co77jugate  iiuaginaries  have  a  real  sum  ayid  a  real 
product. 

For      {x  +yi)  +  {x  —yi) 

=x-j-yi+x—yi,  by  associative  law. 

=x+x+j/z—j/i,  by  commutative  law. 

=  2x+(yi—j/iX  by  associative  law. 

=  2x-\-(j/—y')i,  by  distributive  law. 

=  2x. 
Likewise  (x  +yi') (x—yi) 

=x(x—yz)-\-yi(x—yi),  by  distributive  law. 

=x^'—xyi-\-yzx—yiyi,  by  distributive  law. 

=x'^—-y'^P+xyz — xyz\  by  commutative  law. 
.  =x'^+y'^-\-(xy — xy')z,  by  distributive  law  and 
by  substituting  P  =  —  l.     - 

=x'^-\-y'^. 
It  is  well  to  note  that  l/ze  prod  zed  of  two  co7zjzigate  cozn- 
plex  numbers  is  always  positive  and  the  sum  of  two  squares. 

717.  The  sum,  prodzid,  or  quotient  of  two  coznplex 
numbers  is,  i7z  geiieral,  a  co7nplex  ?zzi7nber  of  the  typical 
for7n  a  +  bi. 

Let  the  two  complex  numbers  be  x+yi  and  zc-[-vi. 

(1)  Their  sum  is  {x-\-yi)  +  {zi-\-vi)    . 

=x-\-yi-{-ti  +  vi, 

=x-\-u-\-yi+vi, 

=  (x-i-u')  +  (y+v)i, 
by  the  laws  of  algebra.     This  last  expression  is  in  the 
form  a-\-bi. 

(2)  Their  product  is  (;t:+j'0(?^ +  2;/) 

=  x(u  +  vi)  -\-yi(zi  +  vi) , 
=  xu + xvi+yiu  -\-yivi^ 
= xu  -i-yvi'^  +  xvi+yui, 
=  (xu  —yv)  4-  (xv  -\-y2c)i, 

by  the  laws  ot  algebra.     This  last  expression  is  in  the 

form  a  +  bi. 


558  UNIVERSITY    ALGEBRA. 

(3)  Their  quotient  is 

X  -\-yi (x  -\-yi  )  (ji — vi) 

u  -f  vi      (ii-[-  vi)  (jc — vi) 
B}^  the  preceding-,  the  numerator  is  of  the  form  a'-^b'i. 
By  Art.  716,  the  denominator  equals  ti'^-\-v'^.     Then  the 
quotient  equals 

a'-\-b'z  ^'      _L       ^'      • 


by  distributive  law.     This  last  expression  is  of  the  form 
a  H-  bi. 

KXAMPI.ES. 

Reduce  the  following  expressions  to  the  typical  form 
a-\-bi:  , 


3.   (;^_[2  +  3z])(x-[2-3/]). 

4.  (_5+l2l/:I:l)^       6.  (v'i+^-)(i/i^*). 

5.  (3«4i/i:i)2.  7.  (i/7_  1/1:7)2^ 

Q         ^  1 

2  1-23 

9.  ;rTT7=^-  ^3 


10. 


3  +  1/--2  (1-0' 

I-1/-7'  ^"^^  l+2i/^* 

_    a+xz     a — xz 

10.    : ; ;• 

a  —  xz     a+xz 


COMPLEX    NUMBERS. 


559 


718.  If  cin  imaginary  is  equal  to  zero,  the  imaginary 
and  real  parts  are  separately  equal  to  zero. 

Suppose  x-^yV  —  1=0 

then  x= — yV —1. 

Now  it  is  absurd  for  a  real  number  to  equal  an  imagin- 
ary, except  they  each  be  zero. 
Therefore  x=0  and  y=0, 

719.  If  two  imaginaries  are  equal,  then  the  real  and  the 
imagi?iary  parts  must  be  respectively  equal. 

For  if  X'\-yi=:^u-\-vi 

then  {x-^ii)-\-(^y—v)i=0. 

Whence,  by  Art.  718, 

x—u-=0  and  j/— z/=0. 
That  is,  x=u  and  y=v. 

MODUIvUS   AND   AMPI^ITUDK. 

720.  lyet  the  complex  number  x-\-yi  be  constructed, 
as  in  figure  19,  in  which  OA=x  and  AP=y.  Draw  the 
line  OP,  and  let  the  angle  A  OP  he  called  0. 

y 

p 


"T 


Fig.  19. 


721.  The  numerical  length  of  OP  is  called  the  Mod- 
ulus of  the  complex  number  x-\-yi.  It  is  algebraically 
represented  by  -\- V  x'^ -\-y^ ,  in  which  the  sign  +  is 
placed  before  the  radical  to  show  that  merely  the  nu- 
merical value  of  the  square  root  is  called  for.  Thus, 
mod(3  +  40= +  1/9  +  16=5. 


560  UNIVERSITY    ALGEBRA. 

The  student  can  easily  see  that  two  conjugate  complex 
numbers  have  the  same  'modulus^  which  is  the  positive  value 
of  the  square  root  of  their  product, 

\iy—^,  the  mod  {x-^yi)  —  \/x^  =  ^x.  Thus  the  modulus  of  any 
real  ^umber  is  the  same  as  what  is  called  the  niunerical  or  absolute 
value  of  the  number.     Thus,  mod  (—5) =5. 

722.  In  Fig.  19  the  angle  A  OP  or  0  is  called  the 
Argument  or  Amplitude  of  the  complex  number  x+yi. 

Putting  r=  +  Vx'^-\-y'^  =  VLioA(^x-\-yi),  we  have 

sin6'=-,  cos^=— 

r  r 

Therefore, 

x+yi=r  cos  0-{-tr  vSin  ^=r(cos  O  +  i  sin  9) 
in  which  we  have  expressed  the  complex  number  x+yi 
in  terms  of  its  modulus  and  amplitude. 
To  put  3—4/  in  this  form,  we  have 


.y^     4-  ^^./3     ^ 


o 


mod  (3-40  =  '/9  +  16=5;  sin  e=^=-~\  cos 

r  b  r      o 

Therefore,  (3-40  =  5(f-4)/. 

The  amplitude  of  all  positive  numbers  is  0,  and  of  all  negative 
numbers  is  180'^.  The  unit  expressed  in  terms  of  its  modulus  and 
amplitude  is  evidently  l(cos  0+^  sin  0). 

723.  The  point  P,  located  by  OA+AP  or  x+yi,  may 
also  be  considered  as  located  by  the  directed  line  OP-, 
that  is,  by  a  line  starting  at  O,  of  length  r  and  making 
an  angle  0  with  the  direction  OX.  A  directed  line,  as 
we  are  now  considering  OP,  is  called  a  Vector.  When 
thus  considered,  the  two  parts  of  the  compound  operator 

r-  (cos  ^+/sin  0)  - 1 
receive    the    following    interpretation:       The    operator 
(cos  0  +  i  sin  ^),  which  depends  upon  0  alone,  turns  the 
unit  through  an  angle  0.     The  operator  r  is  a  tensor, 
which  stretches  the  turned  unit  in  the  ratio  r:  1.     The 


COMPLEX   NUMBERS.  561 

result  of  these  two  operations  is  that  the  point  P  is 
located  r  units  from  6>  in  a  direction  making  the  angle  ^ 
with  OX. 

Thus,  the  operator  (cos  B-\-i  sin  ^)  is  simply  a  more 
general  operator  than  /,  but  of  the  same  kind.  The 
operator  i  turns  a  unit  through  a  right  angle  and  the 
operator  (cos  0-\-i  sin  &)  turns  a  unit  through  an  angle  B. 
If  0  be  put  equal  to  90°,  cos  0-\-i  sin  0  reduces  to  /. 

For  ^=0,        cos  ^+/sin  0  reduces  to      1. 

^=90°,    cos  ^+2*  sin  B  reduces  to       /. 

^=180°,  cos  <9-j-2  sin  B  reduces  to  —1. 

^=270°,  cos  B+i  sin  B  reduces  to  —  /. 

Since  3— 4^=5(f— fz)  the  point  located  by  3  — 4^' may 
be  reached  by  turning  the  unit  an  angle  ^=sin~^(-f) 
=cos~^  f  and  stretching  the  result  in  the  ratio  5:1. 

724.  If  d  complex  numbef  vanishes,  its  modulus  van- 
ishes; and,  conversely,  if  the  modulus  vanishes,  the  complex 
number  vanishes. 

1{  x+yi=0,  then  x=0  andj/=0,  by  Art.  718.  There- 
fore, l/x^-i-y'^=0.  Also,if  y  ^^M^=0,  thenx'^+j;^=0, 
and  since  x  andj/  are  real,  neither  x^  nor  j/^  are  negative,, 
and  so  their  sum  is  not  zero  unless  each  be  zero. 

725.  1/  two  complex  numbers  are  equal,  their  moduli 
are  equal,  but  if  two  moduli  are  equal,  the  complex  7ium- 
bers  are  not  7iecessarily  equal. 

If  x-\-yi=u-\-vi,  then  x^u  and  y-=v  by  Art.  719. 
Therefore,  V x'^^-y'^  =  V u''-'-\-v'^ . 

But  if  l/.a;2+_y2==|/^2_j_^2^  ^2  jieed  not  equal  tc'^  nor 


y'^=zv 


fi-U.  A. 


562  UNIVERSITY   ALGEBRA. 

726.  The  modulus  of  the  sum  of  two  or  more  complex 
niiiubers  is  never  greater  than  the  sum  of  their  moduli. 

If  ^^  =  Xi-j-y^i  and  ^2=-^2+>'2^*  t)e  any  two  complex 
numbers,  we  are  to  prove  that 

mod  (-2'i-f  2'2)>mod -s'l+mod  ^2> 

or+l/(^l+-^2)V(J^l+J^'2)^>  +  ■^■^l^4-JKl^  +  ^/:r2  24-J^^2^ 
but  since  the  square  roots  are  all  positive  it  is  sufficient 
to  prove  that 

or  that 

^1^2 +^1  ^2  >  ^'^1 ' -+-j^ '  X  "^^-^2 ' +JK2  ^ 

or,  again  squaring,  we  must  prove  that 
x^  2^2  ^  +2^1-^2:^1  JK2  +yi  V2  '^ 

>^i  2^2  ^  +-^2  ^^1  ^  +-^1  ^^2  ^  +J^i  Ij^2  ^ 
or  that        0y>X2^yi'^—2xj^X2yiy2+^i^J^2^f 
or  that  0>Cr2jri— ^lJK2)^ 

which  is  obvious,  since  x^,  x^,  yi  and  y^  are  real. 
Likewise, 

mod  (^2^+22+2z)>mod  2^+mod  {s^+^z) 

>mod  ^1  +mod  <2'2  +mod  2:.^, 

727.  The  product  of  tzvo  or  more  complex  numbers  is  a 
complex  number  whose  Tnodulus  is  the  product  of  the  moduli 
and  whose  ajnplitude  is  the  sum  of  the  amplitudes  of  the 
complex  numbers. 

Let  the  complex  numbers  be 

2^=^x^-\-y^i=r-^(sos  0^-\-ism  0^) 
^2=;i;2+jK22'=^'2(cos  ^2+^*  sin  0.f),  etc. 

We  are  to  prove  that  mod  -a' ^-s* 2= mod  2^  mod  2 2, 
and  that  amp  5'i2'.j  =  amp  ^i+amp  2^. 


COMPLEX   NUMBERS.  563 

By  actual  multiplication,  we  get 
2'^^^-rr[(cos^^cos^^— sin^^sin^^)  +  (sin^^cos^^+cos^^sin^J/] 

=  ^iV2[cos(^i+^2)  +  ^*  sin(^i+^2)]. 
Whence  it  is  seen  that  r^  r 2  is  the  modulus  of  the  product 
and  (^1  +  ^2)  is  the  amplitude. 

Also  mod  ^i2'2-s'3=mod  ^1  mod  -3'2'2'3, 

but  mod  ^2'2'3=inod  ^2  ^^d  ^3. 

Therefore/     mod  z^2,^2^=mo6.  2^  mod  22  mod  2^,  etc. 

Likewise  amp  2'i^2'2'3  =  amp  2 ^+2im^ 2^,2^, 
but  amp  ^2-^3==^^? -^2  +  ^^? '^'3. 

Therefore     amp  -3'i^2'2'3=amp  -s'^+amp  ^2+^^P  '2'3,etc. 

728.  The  quotient  of  two  complex  numbers  is  a  complex 
number  whose  modulus  is  the  quotient  of  the  moduli  and 
whose  amplitude  is  the  difference  of  the  amplitudes  of  the 
two  complex  numbers. 

Let  the  complex  numbers  be 

^j=;trj+jKii5*=^i(cos  ^i+/sin  B^ 
22=X2+y2^===^2(^os  O^  +  i sin  O^) 
we  are  to  prove  that 

^^1     mod  2^ 

mod— = :; -y 

22     mod  22 
and  that  amp— =  amp  ^i —amp  . 3*2 

we  have 

^^__ri(cos  ^1  4-2  sin  ^i)(cos  ^2~^*sin  ^2) 
^2     ^2 (cos  ^2+^  sin  ^2)(cos  ^2~^  sin  ^2) 

^r,[cos  (^,-^2)  +  /  sin(^i-6>2)] 
r2(cos2^2+sin2(92)  " 

=^[cos((9i-^2)  +  ^*sin((9i-6>2)], 
^2 

Whence  it  is  seen  that  —  is  the  modulus  ol  the  quotient 

and  (^1—^2)  is  the  amplitude. 


564  UNIVERSITY    ALGEBRA. 

729.     De  Moivre's  Theorem.     As  a  special  case  of 
Art.  727,  consider  the  expression 

(cos  0-^z  sin  Oy 
This  being  the  product  oi  .n  factors  like  (cos  0+z  sin  (?), 
we  write,  by  means  of  Art.  727, 

(cos  0+z  sin  ^)(cos  0-\-i  sin  0)  •  -  - 

=  [cos(<9+(9-|- .  .  .)-f  ^'  sin((9+(9+  .  .  .)] 
or  (cos  ^+2  sin  ^)"=(cos  >^^H-/ sin  ?iO)  [3] 

which  relation  is  known  as  De  Moivre's  theorem. 

DeMoivre's  theorem  holds  for  fractional  values  of  n. 

For,  first  consider  the  expression 

1 
(cos  ^+/sin  oy 

A 

Put  ^=^<3^,  SO  that  <^=-- 

1  1 

Then         (cos  ^+/sin  ^)^  =  (cos  i<fi+i  sin  i<i>)' 

=  [(cos  <t>-\-i  sin  </>)^ '  by  [3] 

=cos  <f>+i  sin  0 

«=cos  -+2  sm  -• 

Next  consider  the  case  in  which  n=-'  We  know 

(cos  0+ i  sin  B) \  =  [(cos  B-^i  sin  ^)^] ' 

«=(cos  ^^  +  /sin^^)' 

sB  ^  ,   ,  sB 
=  cos  — +2  sm-- 

I^ikewise,  the  theorem   may  be  proved  for  negative 
values  of  n. 


COMPLEX    NUMBERS.  565 

KXAMPI.KS. 

1.  Find  value  of  (-1-f  l/^)^  + (-l-l/^)^  by 
De  Moivre's  theorem. 

2.  If  ;tr^=  cos^  +  /  sin^>prove  that  limit  [^i^r 2^3... :r J 
as  r  increases = cos  it. 

3.  Find  the  value  oi x'^—2x-\-2  for  x=\-\-i, 

4.  If  yi  =  — 4-+|^l/+3   and  y2  =  —i— i^/^,    show 

inai  7i   — i,  J2   — ^y  J\   --V2>    72      yi)    yi        72     — -"^j 

y  3«+i — y 


CHAPTER  XXXII. 

TH^  RATIONAI,  INTKGRAI,  FUNCTION. 

730.  Variable  and  Constant.  A  Variable  is  a 
number  or  quantity  existing  under  such  a  law  or  suppo- 
sition that  it  has  an  unlimited  number  of  values.  A 
Constant  is  a  number  or  quantity  existing  under  such  a 
law  or  supposition  that  it  has  a  fixed  value. 

,  Thus  the  volume  of  mercury  in  a  thermometer  is  a  variable  and  its 
weight  or  mass  is  a  constant. 

731.  Function  of  a  Number.  A  Function  of  a 
number  is  a  name  applied  to  any  mathematical  expres- 
sion in  which  the  number  appears.     Thus, 


•^  x^—S  * 

are  all  functions  of  x.  In  the  same  manner  we  speak  of 
functions  of  several  numbers.  The  second  expression 
above  may  be  called  a  function  of  x  and  y.  Obviously, 
a  function  of  a  number  might  be  otherwise  defined  as 
any  expression  which  depends  upon  the  number  for  its 
value. 

732.  Rational  Integral  Function.  A  Rational 
Function  of  a  number  is  one  in  which  the  number  is 
not  involved  in  a  radical  or  is  not  affected  with  a  fractional 
exponent.  An  Integral  Function  of  a  number  is  one 
in  which  the  number  does  not  appear  in  the  denominator 
of  a  fraction  or  is  not  affected  with  a  negative  exponent. 

X  — ^ 

Thus  x^ '-'2x'^y~^  -\-h—^  —  6>/  ^  is  a  rational  function  of  x, 

an  integral  function  of  x,  an  irrational  function  of  jj/,  a 
fractional  function  of  ^. 


RATIONAL   INTEGRAL    FUNCTIONS.  56/ 

A  function  may  be  both  rational  and  integral,  in  which 
case  it  is  called  a  Rational  Integral  Function.  If  n  is 
a  positive  whole  number  and  a^,  a^,  a^y  .  .  .  ««  stand  for 
any  real  numbers  whatever,  then 

is  a  general  expression  representing  any  rational  iJitegral 
function  ofxof  the  nth  degree,  if  we  confine  ourselves  to 
the  case  of  real  coefficients. 

Such  expressions  2,s  function  of  x,  function  of  a,  func- 
tion of  x-\rh,  etc.,  are  abbreviated  into  F{x),  F(a), 
F{x+h)y  or  fix),  f (a),  f(^x+h),  or  a  similar  expression. 
It  must  be  kept  well  in  mind  that  F,  f  etc.,  are  not 
coefficients,  but  abbreviations  for  the  words  ''function  of.'* 

If/(;r)  and/(^),  or  7^(^)  and  F{a),  occur  together  in 
the  same  discussion,  f{a)  stands  for  what  /(jt)  becomes 
when  a  is  put  for  x.  Thus,  \^  f{x)  is  x^—lx'^-{-2x-\-4:, 
then/(a)is  ^'^—T^^  ^2^+4, and /(2)  is  2=^  --7  -22  +  2  -2  +  4 
or  —12.  In  the  vSame  way, /(;»;+/2)  stands  for  what/(jr) 
becomes  when  {^x-\-h)  is  put  for  x. 

733.  Notation.  The  symbol  /(jr)  will  be  used 
throughout  this  chapter  to  stand  for  a  rational  integral 
function  of  x.  A  function  which  is  not  both  rational  and 
integral  must  be  represented  by  one  of  the  symbols  F{x), 
^{x^y  etc. 

If  we  suppose  f{x^  to  be  divided  through  by  the  coeffi- 
cient of  the  highest  power  of  x,  then  the  following  will 
represent  any  fix) : 

If  none  of  the  above  coefficients  are  zero,  the  function 
is  said  to  be  Complete.  The  term  ^„,  or  />„,  is  called 
the  Absolute  Term. 


568  UNIVERSITY    ALGEBRA. 

734.  Continuous  and  Discontinuous.  A  Contin- 
uous Variable,  in  passing  from  one  value  to  another, 
passes  over  every  intermediate  value.  Thus,  if  x^  in 
passing  from  2  to  7,  passes  over  every  intermediate  value, 
then  X  is  said  to  be  continuous  between  the  limits  2 
and  7.  The  repeating  decimal  .33333-  •  .  is  an  illustra- 
tion of  a  Discontinuous  Variable,  as  in  passing  from 
.33  to  .333333  it  skips  all  intermediate  values  except 
.333,  .3333,  and  .33333. 

735.  Identities.  Two  expressions  that  are  equal 
for  all  values  of  the  letters  involved  are  said  to  be  Iden- 
tically Equal  or  Identical.  See  Art.  214.  The  symbol 
for  identity  is  =.  Thus,  we  write  (x-f<2)(^—^)  =  ^2—<22. 
A  very  convenient  use  of  the  symbol  =  is,  in  writing 
''let  /(x')  =  x^—4:ax  +  a'^j''  instead  of  "let  /(x)  stand  for 
x'^-4ax-{-a\'' 

736.  Roots.  Any  real  number  or  imaginary,  which 
substituted  for  x  in/(x)  makes /(:r)  vanish  (that  is,  equal 
to  zero)  we  shall  call  a  Root  of /(:r).  Thus,  1,  2,  and  3, 
are  roots  of  x^  —  6x'^  +11x^6  and  2, 2+3/,  2— 3/ are  roots 
of^3-6:r2-5;r-26. 

737.  A  Rational  Integral  Equation  containing  one 
unknown  number  is  one  which  can  be  placed  in  the  form 
/(;t)=0;  that  is,  in  the  form 

af)X"-ha^x''-'^  -{-a^x"-'^  +a^x''-^  ^  •  •  • +a,,_i^+a„=0.  (1) 

A  rational  integral  equation  may  also  be  represented  by 

x^+p,x'^-^  4-^2^-2  +^3^«-3  +  .  .  .  +^_^^+^^ -0    (2) 

since  the  equation  (1)  is  unchanged  if  we  divide  through 
by  the  coefficient  of  x"".  When  the  equation  /(x)  =  0  is 
written  out  in  either  of  above  forms  it  is  commonly  spoken 
of  as  the  General  Equation  of  the  7zth  degree. 


RATIONAL   INTEGRAL   FUNCTIONS.  569 

"  738.  The  difference  between  f{x)  and  f(a)  is  exactly 
divisible  by  x—a. 

We  are  to  prove  -^—^ — ^^-^  =  a  quotient  without  a  re- 
mainder. 

Now    f(x)  =  aQX"  +  a^x''~'^+aci^x''~'^+  -  •  •  +a„_^x+a„ 
and       y*(a)  =  (2Qa"  +  ^ia"~i+<^2^"~^+  •  •  •  +<^«-iOt+^«. 

Therefore,  -^  ^  ^    -^  ^  ^= 

X — a 

Jf— a 

equals  some  quotient  without  a  remainder,  since  the  dif- 
ference of  like  powers  is  divisible  by  the  difference  of  the 
numbers  themselves,  by  Art.  136. 

739.  When  fix)  is  divided  by  x—a,  the  first  remainder 
that  does  not  contain  x  is  equal  to  /(a). 

As  an  illustration,  divide  x^ — ^x'^+llx—Q  by  x—a, 
as  follows : 


x^—6x^  +  llx-6 


X 


3-_n 


X—a 


^2  +  (a~6>"+(a2-6a-fll) 


(a—6)x'^  —  (a'^—6a)x 

(a2-6a+ll>-6 
•       (a2-^6a+ll).r-(a^— 6a2  +  lla) 
"  a3_-.6a2  +  lla— 6 

Thus,  the  first  remainder  which  does  not  contain  x  is 
1^— Ga^H-lla— 6,  which  can  be  made  from  x'^  —  ^x'^  +  llx 
—  6  by  putting  a  for  x. 

TTo  prove  the  theorem,  we  have,  from  Art.  738, 
fix) -/{.-)_ 


x—a 


-  =  quotient,  no  remainder. 


That  is,  A^_)  _/(.«)_  =q,,otient. 

X—a       X—a        ^ 

Therefore  -^-^  =  quotient  +*^-^ 

X — a  X — a 

Therefore  /(a)  is  the  remainder. 


570  UNIVERSITY    ALGEBRA. 

740.  Any  fix)  is  exactly  divisible  by  x  mi7iiis  a  root, . 
Let  the  root  be  a. 

Now  from  Art.  739/-^-^^ — ^^-^— =  quotient,  no  remainder. 

X — a 

But  since  a  is  supposed  to  be  a  root,y*(a)=0  by  Art.  736. 
Therefore, ^=  quotient,  no  remainder. 

We  know  1,  2,  and  3  are  roots  of  x^  —^x^ -\-\\x — 6.  Therefore, 
by  this  theorem,  x^  —^x^-\-\\x-^  is  exactly  divisible  by  Jt  — 1,  x  —  2, 
and  x  —  ^. 

741.  Conversely,  if  xny  f(x)  is  exactly  divisible  by 
X — a,  then  a  is  a  root  of  f{x). 

Since  f{x)  is  exactly  divisible  by  x—a,  we  know  that 
x—a.  is  one  factor  of/(;t:).  Representing  the  other  factor 
by  <^(;r),  we  have 

f{x)~{x—a)  <t>(x). 
Substituting  a  for  x,  we  get 

/(a)  =  (a-a)</>(a)  =  0. 

Since  f(x)  becomes  zero  when  a  is  put  for  :r,  a  is  a  root 
of  /(;r)  by  definition. 

742.  The  above  theorem  is  sometimes  useful  for  indi- 
cating the  factors  of  an  expression,  as  in  the  following 
examples: 

(1).     Factor  {d-c){d-\-c)^ +  {c—a){c-\-ay -\-{a—l>){a-\-d)^. 
Put^=<:.    The  expression  becomes  (^  —  rt!)(<:+«)2  +  («--^){rt!4-^)^,  or  0. 
Since  the  original  expression  vanishes  for  d:=^c,  it    is   by  Art.   740, 
divisible  by  ^—c.     Likewise  it  will  be  found  to  be  divisible  by  c-^a 
and  d  —  a.     Whence,  we  have 

Since  each  side  of  this  identity  is  of  the  third  degree,  Z  is  an  undeter- 
mined number  independent  of  a,  b,  and  c.  On  the  left  side  the  coeffi- 
cient of  b'^c  is  —1.  On  the  right  side  the  coefficient  of  b'^c  is  — Z. 
Therefore.  Z=:l. 


RATIONAL   INTEGRAL   FUNCTIONS.  57 1 

(2).     Factor  {d-c)^-\- {c-a)^-t {a- d)^. 

This  expression  is  zero  for  d=c,  c—a,  a—b.  Therefore,  by  Art. 
740,  it  is  divisible  by  {b—c){c—a){a—b).  But  the  expression  being  of 
the  fifth  degree  it  must  have  a  remaining  factor  of  the  second  degree, 
which  maybe  represented  by  La^-{-Lb^ ■\-Lc'^-\-Mbc-\- Mca-\-AIab,  in 
which  Z«and  J/ are  undetermined  coefficients.  Therefore,  we  have 
{b—a)^  +  {c—a)^-{-{a-  b)^ 

={b—c){c—a){a-b){La^-^Lb^-\-Lc'^  +  Mbc  +  McaA-Mab). 
The  coefficients  of  «*  on  the  respective  sides  of  this  identity  are 
seen  to  be  given  in 

{^c-bb)a^--L{b-c)a'^ 
Whence  Z=:4-5. 

The  coefficients  of  b^c^  are  likewise  seen  to  be  given  in 
+  lQb^c^  =  {-M+L)b^c^, 
whence  —M-^tL^^O,  or  M=  —  ^. 

Therefore  the  factors  of  the  original  expression  are 

b{b-c){c—a){a-b){c^  +  b'^  +  c^  —  bc-ca—ab). 

EXAMPLES. 

Show  by  Art.  740  that  the  following  expressions  are 
divisible  by  (b—c)(c—a)(a — b).  Remaining  factors  may 
be  found  by  the  principle  of  undetermined  coefficients,  if 
desired. 

1.  bc{b—c)  +  ca(c—d)  +  ab(a—b), 

2.  (^b—cy  -{-(c—ay  -^{a—by . 

3.  (b—c)(a — b+c)(a-{-b—c)-^(c—-d)(a^b—c)(^-a-\-b-\-c) 

+  (a--b){'-a-\-b^c){a  —  b+c). 

4.  a^{b-c)-^b^(c—a)  +  c^{a—b). 

5.  a{^b-cy  ^b{c-ay  +c(a-by\ 

6.  (^b-'C)(ib-\-cy  +  (c-a)(ic+ay-^{a-^b){a  +  by. 

7.  b''c''(ib-c)-\-c''a''(jr-a)  +  a''b''(a-b), 

8.  a^{b—c')-^b\c-a)-{-c^(^a—b). 

9.  {b-cXb-^-cY^^c-a^tic+ay  +  ^a-bXa  +  by. 


572  UNIVERSITY   ALGEBRA. 

10.  a(d—c)^  +  5(c—a)^+c(a'-d')^. 

11.  a^(d—c)-i-d^(c--a)  +  c^(ia—d). 

12.  d^ c^ (d—c)  +  c^ a^ (c—a)  +  a^ d^  (a  —  d"). 

Show,  by  Art,  740,  that  the  following  expressions  are 
divisible  by  (<^+r)(<;+<^)(<^  +  ^): 

13.  (a  +  d+c)^—a^  —  d^ — c^, 

14.  (a  +  d+c)^—a^—d^—c^. 

15.  Show  that  a4(^^-^')  +  <^^(^^-^^)+^^(^'-~^^) 

16.  Show  that  (a  +  d+cy  —  (id  +  c-'ay  —  (ic+a-d)^ 
—  (a-{-b—c)^  is  divisible  by  abc, 

743.  Synthetic  Division.  We  shall  now  explain 
a  short  method  of  dividing  any  f(^x)  by  x—a.  For  con- 
venience suppose  the  f{x)  to  be 

for  it  will  be  plain  that  the  process  to  be  explained  will 
be  applicable  to  a  polynomial  of  any  degree  whatever. 
It  is  known  that  the  quotient  oi  f{x)-h-{x—a)  will  be  of 
one  lower  degree  than  /(.r),  therefore  we  may  assume. 
a^x^  ■\-a^x^-{-a^x^-\-a^x'^-\-a^x-\-a^ 
X — a 

X — a 

in  which  A,  B,   C,  etc.,  are  undetermined  coefficients. 
Put  R  for  Remairider.     Then 

a^x^  '\-a^x^-\-a.^x^  +a^x'^-\-a^x+a^ 

=  {x--a\{Ax^  +  Bx^  +  Cx-  -\-Dx  +  E  +  -^  \ 
^Ax^+(B—aA')x'-j-(iC-aB)x^ 

+  iiD'-aC)x'+(£'-aD)x+R-aB, 


RATIONAL   INTEGRAL   FUNCTIONS.  573 

Equating  coefficients  of  like  powers  of  x,  by  Art.  595, 
we  get 

A  =^0  A=  a^ 

B—aA=a^  B=aA-{-a^ 

C-aB^a^        Therefore  C=^aB+a^ 

Z7-aC=«3        ^^eretore,  D=aC+a^ 

R—aE=a^  R=aE-\-a^ 

Arranging  the  right  hand  column  of  equations  horizontally : 
Coefficients 

in  dividend,    a^     -f^^      +^2      +^3     +^4      +(1^ 

+  aA       -^aB      4-aC      -^aP      -\-aE 

Coefficients  ||  ||  ||  ~||  ||  || 

in  quotient,  A  B  C  D         E  R 

From  this  we  observe  the  following  law  of  coefficients 
in  the  quotient: 

The  first  coefficiejit  in  the  quotient  is  the  same  as  the  first 
coefificient  in  the  dividend.  The  second  coefficient  i7i  the 
quotieyit  equals  the  second  coefficient  in  the  dividend,  plus  a 
times  A,  the  one  just  fo2ind.  The  third  coefficient  in  the 
qjwtient  equals  the  third  coefficient  in  the  dividend  plus  a  times 
By  the  one  just  found.  And  any  coefficient  in  the  quotient 
equals  the  correspo7iding  coefficient  in  the  dividend  plus  a  times 
the  preceding  one  in  the  quotient. 

The  process  of  finding  the  coefficients  in  the  quotient, 
and  the  remainder,  is  more  apparent  in  particular  cases  : 

(1).  Find  the  quotient  of  Ix^+^x'^-^x^ -\^x^—%x-{-^  by  ^-2* 
Coefficients  in  dividend,  7      +8—6      —15—8      +4     (2 

14      44         76     122     228 

Coefficients  in  quotient,  7       22       38         61     114     232 

Hence  the  quotientis,  7;*r4  +  22jr3  +  38;ir2  +  61x  +  114  +  -^ 

x—% 
(2).   Find  the  quotient  of        x^  —  '^\  by  ^-3: 
Coefficients  in  dividend,  ,1         0         0        0      -81      (3 

3         9      27         81 
Coefficients  in  quotient,  1         3  9       27  0 

Quotient,  x3  +  3;t^  +  9;«^+27,  no  remainder. 


574  UNIVERSITY   ALGEBRA. 

This  method  of  obtaining  a  quotient  is  called  syjithetic 
division.  It  will  evidently  apply  whatever  the  degree  of 
the  dividend. 

What  change  will  there  be  if  the  divisor  is  ;^;  +  a? 

744.  The  short  method  of  division  together  with  the 
theorem  of  Art.  739  furnishes  a  short  way  of  finding  the 
value  of  2iViy  f{x)  for  a  given  value  of  its  variable.  Thus, 
suppose  we  wish  the  value  of  x^ ^^ x^ -^-^x'^  +  Vdx-\-*l^ 
when  x=h\  that  is,  we  wish/(5).  Now,  by  Art.  739, 
the  remainder  when  /(x)  is  divided  by  x—6  is  /(5). 
Whence, 

1         -7         +6         4-10         4-70     (5 

5       -10         -20 -50 

1         -2         -4         -10         4-20 
That   is,    the   value   of  ;i:*— 7:^3  4- 6.r 24- 10.^4-70    when 
x=h  is  20.     Such  examples  should  be  done  in  this  way, 
as  it  is  a  shorter  process  than  direct  substitution. 

KXAMPI.KS. 

1.  Divide^t:*— 5;»;3  4-12;r2  4-4;t:— 8by  ;»:— 2. 

2.  Divide:tr3  4-ll-^^  +  36;t:4-15by  .r4-5. 

3.  Divide  ;r^ 4- 6;t:4-10;i:3-112;i;2-207.r-110  by  ,v4-5. 

4.  Divide^i;*  — 12;r3  4-47:r2— 72ji;4-36by  ;r— 5. 

5.  Prove  that  3  is  a  root  oi x^--^x'^  +  llx—^.    Art.  741. 

6.  Prove  that  6  is  a  root  of  ;r*  —  12;r3  +  47ji:2— 72^4-36. 

7.  Prove  that  4  is  a  root  of  .^i;*— 55;f2  4-30;r4-'504. 

8.  Prove  that  —11   is  a  root  of  x^-^lOx^'-Zbx'^—ZOOx 

-396. 

9.  Provethat— 6isarootof;i:*— 4.r3— 29.^-2  4.l56;^;— 180. 


RATIONAL    INTEGRAL    FUNCTIONS.  575 

10.  mndva\ueo^x^  +  16x'^—ix+li{x=2.     Art.  739. 

11.  Find  value  of  jr^  +  9;t:2  —  19;r--76  when  :r=  5. 

12.  Find  value  oi  x^  —  Sx'^ +ox-{-178  when  x=^4. 

13.  Show  that  (l  +  a)\l+c^)-(l  +  dy(l+c^)  is  ex- 
actly divisible  by  a—d.     Art.  738. 

14.  Show  that/(«2)— /(/^^)  J3  exactly  divisible  hya  +  d. 

745.  /f  ^  is  any  complex  number^  then  f(£)  is  a  com- 
plex fiicmber. 

Lfet/(^)  =  aQ2:''+a^3''"'^-\-  •  •  •  +a,^_-^2:+a„2ind2=x+yi. 
Now  in  the  expression 

^o(-^+j'0"+^i(-^+j^O""^+  •  •  •  +^«-i(-^+j^O+^«  (1) 

let  the  terms  (jt+jkO")  C-^+J^O"""^)  etc.,  be  expanded  by  the 
binomial  theorem.  Then  all  the  terms  containing  yi  to 
an  odd  power  will  be  imaginary,  and  their  sum  may  be 
represented  by  Vz.  All  other  terms  will  be  real  and  their 
sum  may  be  represented  by  X,     Thus,  we  have 

746.  ///ix+yi')  =  X+  Vz,  lken/(x-yz)  =  X-  Vz, 
For,  changing  the  sign  of  y  will  change  the  signs  of 

all  the  odd  powers  of  yz,  and  hence  V  will  be  changed 
in  sign,  but  not  otherwise  altered.  X,  being  the  sum  of 
terms  independent  of  y  and  of  terms  containing  only  even 
powers  of  yz,  will  not  be  affected  by  changing  the  sign 
of  y.     Hence,  if 

f(x+yz)  =  X-^Vz 
then  A^—y  0  =  ^—  ^i 

747.  To  Express  f(x+h)  in  Powers  of  h.  Suppose 

f{pc)  =  aQx''+a-^x''~'^+a2x"~'^-\-  -  -  •  +«„_i^+^«. 
Then  we  have 

/(X'^k)  =  ao(x+/iy+a^(x-{-/iT''^ 

-^a^(ix+ky-^+  .  .    +a,,_^(x+/i')+a„. 


576  UNIVERSITY   ALGEBRA. 

Expanding  (x+k)",  (x-}-ky'~'^,  etc.,  by  the  binomial 
theorem,  and  collecting  in  terms  of  powers  of  k,  we  have 
aQx"  +  aiX"-^+a^x"-'^+  •  •  •  +a,,_-^x+a,, 
+  /i[?iaQX"-'^  +  (n—l')a^x"-^  +  (i7i—2)a.2x''-^+  •  .  .' 

-^^[?i(i7i-l)a^x"-''  +  (in-'l)(7i-2')a^x"-^ 

+  (/^~2)(;^-3>o-r«-4+  .  .  .  +2a,_2] 
+  •  •  • 

It  is  observed  that  the  portion  of  the  result  independent 
of  h  isy(;i;),  and  that  the  coefficients  of  the  successive 
powers  of  k  are  functions  of  x  of  degrees  decreasing  by 
unity.  These  coefficients  are  known  respectively  as  the 
First  Derivative,  Second  Derivative,  etc.,  of/(.:r)  and 
are  represented  by  the  symbols  f'{x),  f'\^)^  etc.,  or 
the  symbols  /i(^),  f<i{x),  etc. 

It  is  important  to  note  that  the  iirst  derivative  may  be 
derived  from /(;»;).  by  the  following  process:  Multiply  eack 
term  hi  f{pc)  by  the  exponent  of  x  in  that  term  and  decrease 
the  exp07ient  in  that  term  by  unity.  Applying  this  same 
process  to  f'ipc)  will  give  us  the  second  derivative  oif(x^, 
and  so  on. 

Thus,  in  symbols,  we  have 

/(^-+/^)=/(^)-f-/'(^)/^+/"W^+  •  •  •  +^0^^".     [1] 
It  is  plain  that  if  z  and  h  be  any  two  complex  numbers, 
the  same  process  applied  to  f{z-\-h^  will  give 

/(^+>^)-/(^)+/'(^)/^+/"(^)j^+  •  •  •  +a,h\      [2] 
Since  x-^h—h-^-x,  we  know  f(x-\- h')=f{h  +  x).    Then 
if  we  interchange  x  and  h  in  the  right  member  of  [1],  we 
obtain  2 

/(^+/^)=/(;^)+/'(/^)^..;-/-(/0J72+  •  •  •  +^o^">     [3] 
in  which  f{x-\-K)  is  expressed  in  powers  of  .r. 


RATIONAL   INTEGRAL    FUNCTIONS.  57/ 

(1).  Find  the  result  of  substituting  x-\-/i  ior  x  in  x^  -  Qx^  +llx—Q. 
/{x)=x^-Qx^  +  11^-6 
/'{x)=3x^-12x+n 

/"(;^)=G^'-12 
/"'{x)=Q. 

Therefore.  /{x^/i)=/{x)-\-/' {x)/i+/"{x)y-^^-{-/'  "{x)t^ 

=x^-Qx'^-j-nx-Q  +  {^x^-12x-\-U)/i+{Qx-12/^-\-Qk^. 

(2).   Find  the  result  of  substituting  :r+5  iovx'm  jt*  — 6^2  +  20x;- 14. 
f{x)^x'^  —  Qx'^  +  20x-l4: 
f'[x)=ix^-\2x-^2Q 

f"{x)=\2x'^-l2 

/"'{X)  =  24:X 

/iy[x)=24:. 
Therefore.     /{x-\-5)=/{x)+/'  {x)5+/"{xy%^-]-/'  "(x)H^-l-/iv(x)-Vf- 
=x^  +  20x3-\-U4.x^-hi(j0x+5Gl. 

•  748.  Continuity  of  f(oo).  If  x  changes  from  the 
value  a  to  the  value  d,  we  wish  to  show  that /(a)  changes 
to  /(d)  by  passing  over  all  the  intermediate  values.  That 
is,  we  wish  to  show  that  /(x)  is  a  co7iti7iuotcs  variable, 
and  not  a  variable  that  skips  values  like  the  variable 
.3333.  .  .  or  .27272727.  •  •,  etc. 

Let  a  be  any  value  of  ;t:,  and  let  a-\-h  be  another  value 
of  X,  Thus,  f{a)  is  one  value  of  f{pc)  and  f(a+h)  is 
another  value  of /(;r).     Then  we  have 

Whence, 

f^a-Vh')--f{a^^f\ayi+f\d)^^-^  .  .  .  j^a,h\ 

Now,  the  right  hand  member  of  this  equation  may  be 
made  as  small  as  we  please  by  taking  h  small  enough, 
since  every  term  contains  h.  Thus,  the  difference  be- 
tween f(a-\-Ii)  and /(a)  can  be  made  as  small  as  we  please. . 
But  a  and  a-\-h  are  two  values  of  ;f  an^fia-^h)  2nd/(«)) 

37  —  u.  A. 


573  UNIVERSITY   ALGEBRA. 

are  two  values  oi  f{x).  Thus  we  have  shown  that  the 
difference  between  successive  values  of  f(x)  can  be  made 
as  small  as  we  please ;  that  is,  /(x)  is  continuous. 

749.  A7iy  term  in  f{x)  can  be  made  greater  than  the 
sum  of  all  the  terms  of  lower  degree  if  x  be  sufficiejitly 
increased,  and  any  term  in  fix)  can  be  made  greater  than 
the  stem  of  all  the  terms  of  higher  degree  if  x  be  sufficiently 
diminished. 

Let  f{x)^a^x''-\-a^x''~'^^..  ,  +  a^''~''+,  ,,+a„^^x+a*'. 
We  shall  show  that 

if  X  be  taken  large  enough.  Since  x  is  to  be  taken  large, 
^«-^-i  ig  ^\^Q  greatest  of  the  numbers  x"~''~'^,  x'"*"^^  etc. 
Let  a^  be  the  greatest  of  the  coefficients  a^-^i,  ^^+2*  •  •  * 
««_i,  a„.  Then,  since  there  are  ;z--r  terms  in  the  right 
member  of  (1),  we  may  write 

(n^r)asX''-''-'^>  a^+ix"-''-'^  +  •  •  •  +a„_-^x+a„.     (2) 
Thus,  it  sufficient  to  prove  that 

^^"-^>  (n — r^a^''-*"  ^ ,  (3) 

for  much  more,  then,  will  the  left  member  of  (1)  be  greater 
than  the  right  member.  But  dividing  the  members  of 
(3)  by  x"-"-^,  we  have 

a^>(n'-r)asy  (4) 

which  is  true  if  you  take 

a, 

f-^ f^  d 

Thus,  if  you  take  x>-^ ^—^y   the  term  a^*'-''  will  be 

a, 

greater  than  the  sum  of  all  the  terms  of  lower  degree. 
Let  us  next  show  that 

a,x''-''>a^x"+a^x''-''  +  •  •  •  ^-a,_l^''-''+^  (6) 


RATIONAL   INTEGRAL   FUNCTIONS.  379 

if  X  be  taken  small  enough.     Since  x  is  to  be  made  very 
small,  x"~''^'^  is  the  greatest  of  the  numbers  x"",  x"~'^ , .  •  . 

Let  a^  be  the  greatest  of  the  coefficients  ^o>  ^i^  '  •  *  ^r-i- 
Then,  since  there  are  r  terms  in  the  right  member  of  (6), 
we  may  write 

ra,x"-''+^>aQX*'  +  aiX*'-'^  + f-^r-i-^""'"^^ .     (7) 

Thus,  it  is  sufficient  to  prove  that 

a^"-''>ra^""'-^^  (8) 

Dividing  the  members  of  (8)  by  x*'~'',  we  have 

a^^ra^x  (9) 

which  is  true  if  you  take 

Thus,  if  you  take  ;r<— ^>  the  term  «^"~''  will  be 
greater  than  the  sum  of  all  the  terms  of  higher  degree. 

750.  If  2  a7id  h  are  a7iy  two  complex  numbers,  h  may 
be  giveji  such  a  value  that 

mod  f{z-\-li)  <  mod  fiz), 
provided,  of  course,  that  mod  f{2)^0. 

We  know  that 

/(^+/0==/(^)  +/'(^)/i+/"(^)j;^+  •  ■  •  +a,h".       (1) 

In  this  equation,  any  of  the  functions  /'(^),  f'\2),  etc. 
may  be  zero,  but  a^h''  is  not  zero,  since  a^  and  h  are  not 
zero. 

Dividing  both  sides  of  (1)  hyf{z),  we  have 

Smce  2'  IS  a  complex  number,  •^-^^-e^)-^^)  etc.,  are 
complex  numbers  by  Arts.  745  and  717. 


58o  UNIVERSITY    ALGEBRA. 

Representing  these  by  ^^(cos  O^-^-z  sin  ^i),  etc.,  and  k 
by  p(cos  a-f  ^  sin  a),  we  get 

r.  .    =1  +  ^1  (cos  ^1 4-2*  sin  ^i)p(cos  a+/sin  a) 
/  w 

+  r2(cos  ^2  +  ^*  sin  ^2)p^(cos  2a+z  sin  2a) 

+  .  .  . 

+  r„(cos  0„-\-z  sin  ^„)p"(cos  ?ia+z  sin  no),     (3) 
in  which  p^(cos  2a+?  sin  2a),  etc.,  are  written  for  /z^,  etc., 
by  De  Moivre's  theorem.     Any  of  the  moduli  r^,  rg,  etc., 
may  be  zero,  except  r„  is  not  zero. 
By  Art.  727,  we  write 

^j^=l  +  ^l/>   [C0s(^,-|-a)  +  2Sin(^,+a)] 

+  r2p2[cos((92  +  2a)+2sin(i92  4-2a)] 

4- .  .  . 

+  /  „  p"  [cos(^,, + na)  +  z  sin (0,, + ;za)]  (4) 

Since  ^,  and  therefore  p  and  a,  is  at  our  disposal,  take 

a  so   that   ^i+a=180°;    but   if   ri=0,  take   a   so   that 

^2  +  2a=180°,  and  so  on.     If  ?i=0  note  that  all  of  the 

numbers  r^^,  rg,  •  •  •  are  not  zero,  for  r,,^0. 

Then,  taking  ^iH-a=180°  and  letting  the  resulting 
values  ot  ^2+2a,  ^g  +  Sa,  etc.,  be  represented  by  <^2»  S^3> 
etc.,  we  have 

•^-^--^=l--;iP-f^2p^[cos  ^2  +  ^'sin  <;f>2] 
+  ^3,o^[cos  <^3  +  /sin  t^g] 

+ •  • 

+  r„  p"  [cos  <^„  +  z  sin  <;(>  J  (5) 

in  which  p  is  still  at  our  disposal.  Take  p  so  small  that 
i— r^p  is  positive;  that  is,  take  p<— » then  mod  (1— z'lP) 
=1— r,p.     Hence,  by  Art.  726,  we  derive 


RATIONAL   INTEGRAL   FUNCTIONS.  58 1 

Now,  by  Art.  749,  r^p  can  be  made  greater  than  tbe 
sum  of  all  the  terms  of  higher  degree  by  taking  p  small 
enough. 

Then  (6)  becomes 

mod/(2') 

in  which  ^  is  a  positive  number. 

/TA1       r  mod/(^4-/^)    .-, 

Therefore,  • ^r7rT^<^- 

mod/(^) 

That  is,  mod/(2'+//)<mod/(-2'). 

Thus  we  have  shown  mod /(2:  +  y^)<mod  /{z]  by  taking   0^-{-a 

=180^  and /3< ^ .     This  is  on  the  supposition  that  r^^O.     If 

{n-l)rs 
rt  is  the  first  of  the  numbers  ^1,  ^2-  ^'3  '  *  *  ^^^^   ^^^^  '^'^^  vanish,  we 
must  take 

^,+  /a=180o 

and  p<  ^^ 


{n-t-\)rs 


751.  Every  /(x)  has  at  least  one  root,  real  or  imaginary . 
This  follows  immediately  from  the  last  article.     For,  if 

possible,  suppose  that  the  least  value  of  mod  f{pc)  is  not 
zero,  but  let  some  number  ;;^,  be  the  minimum  value  of 
mod/(ji:).  Then  h  may  be  so  taken  that  mod/(jr+/^) 
<mody*(j\;);  that  is,  x  maybe  so  changed  as  to  make 
mod  fix)  less  than  its  minimum  value,  which  is  absurd. 
Therefore,  the  minimum  value  of  mod/(jt:)  can  be  nothing 
else  than  zero.  But  when  mod  /(jr)=0, /(^)=0  also,  by 
Art.  724.  Since  f{x)  has  0  for  one  of  its  values,  it  has  a 
root. 

752.  Any  fipc)  Of  the  nth  degree  has  n  roots  aiid  no 
more. 

Let  the  function  be 
a^x'''^a^x''~^+a^x"~'^-\-  -  -  •  +a„_2^''^+^u^-i^+a»* 


582  UNIVERSITY   ALGEBRA. 

This  has  one  root,  real  or  imaginary  (Art.  751) ;  call 
that  root  a^.     Then  f{j)c)  is  exactly  divisible  by  ;i;— a^ 
(Art.  740).     The  quotient  is  of  the  form 
^o^""^  +  ^i-^""^+-^2-^"~^+  •  •  • 

in  which  A^,  A^,  etc.,  can  be  determined  by  the  law  of 
coefficients  given  in  the  short  method  ot  division.  Hence, 
we  may  write 

/(jt:)  =  (:r-aO[^o-^""^  +  (^^--l)terms]  (2) 

But  the  second  factor  in  this  expression  is  a  rational 
integral  function  of  x  of  the  (72— l)st  degree,  and  there- 
fore, by  Art.  751,  has  at  least  one  root.  Call  this  root  a2. 
Then  this  second  factor  is  exactly  divisible  by  x—a^,  and 
the  quotient  is  of  the  n  —  ^  degree,  and  contains  {^n—V) 
terms. 

Hence,  we  write 

/(^)  =  (:r-ai)(;r-a2)[ao;r"-2^(;2-2)terms]       (3) 

Again,  the  third  factor  ot  this  must  have  a  root,  sup- 
pose a 3 ;  therefore, 

f{pc)  =  {x'--a.^{x—G.<^{x--a^\a^x^'~'^  +  (^2— 3)terms]  (4) 
and  so  on.     Finally,  we  may  write 

/(;i;)~(^-ai)(;i;-a2)  +  ...  +  (:r-aJ(ao;r«-'*-fO  terms)  (5) 
or        f{x)'^{x-'(i^){x—(x^){x  —  o.^^.  .    (:r— aJ^Q^o         (6) 

It  is  obvious  from  (6)  that  /{pc)  will  vanish  when  any 
one  of  the  n  numbers  a^,  ag,  ag,  etc.,  is  put  for  x,  and 
therefore  these  numbers  are  n  roots  of  /(;r).  It  is  also 
evident  if  any  number  be  put  for  x  except  a^  or  ag,  etc., 
that  fix)  will  not  vanish,  and  therefore  f{pc)  has  no  more 
than  71  roots. 

753,  In  consequence  of  the  above,  we  are  at  liberty 
to  use  either  of  the  following  expressions  to  represent 
any/(;r): 

or  {x—o.^{x-'(k.^{x—o.^'  '  •(x—(i„)aQ. 


RATIONAL    INTEGRAL    FUNCTIONS.  583 

754.  Multiplying  or  dividing  f(x)  by  a  constant  does 
not  affect  the  values  of  the  roots. 

Let  p  and  q  be  any  constants:     Then 

All  01  these  functions  obviously  vanish  for  the  same 
values  of  x. 

755.  The  imaginary  roots  of  f{x),  if  any,  occur  in 
conjiigate  pairs. 

Suppose  a+j8/  is  an  imaginary  root  oi  f{x)\  we  shall 
prove  a— pi  is  also  a  root. 

We    know,   by   Art.    740.    that  f{x)   is  divisible  by 

x—a—pi.     If /(^)  is  divisible  by  x—a-i-/3i  also,  then  by 

by  Art.  741,  a—/3i  is  a  root.  Let  us  divide /(^)  by 
(:r— a— /30(;t:— a4-i50oi'bytheequalofthis[(.r— a)2-f/52]^ 

If  there  be  a  remainder,  stop  the  division  when  the  re- 
mainder is  of  the  first  degree,  and  represent  it  by  Sx+  7, 
Then  we  have 

f(^x')  =  Q[(x^ay+l3']  +  Sx+T,  (1) 

Put  x=a-\-l3i,  and  we  have 

f(^o.+l3i)~Ql-l3'^+l3^^-]-\-Sa-^SI3i+T.  (2) 

But,  since  a-\-/3i  is  a  root,  /(a+^/)  =  0.  Therefore, 
since  QI—P2+P2}  is  also  zero, 

Sa+Spi-\-T=0.  (3) 

Here  we  have  an  imaginary  equal  to  zero.  Therefore, 
by  Art.  718,  the  imaginary  and  real  parts  are  separately 
equal  to  zero,  so  that 

S/3i=0. 
Since  ^5^0,  6*  must  equal  zero.     Substituting  kS=0  in  (3), 


584  UNIVERSITY   ALGEBRA. 

Now,  since  5  and  T  are  each  zero,  there  is  no  remain- 
der when/(;i:)  is  divided  by  (:r— a+^2),  and  hence  a—^i 
is  a  root. 

This  proposition  may  also  be  proved  by  Art.  746.  For, 
if /(a-f  j82")=0,  we  have 

/(a+^/)=;^+r2=o. 

Since  X-\-  Vi=-0,  X=0  and  V=0  by  Art.  718.  '  Hence 
X-r^=0  also.  But,  by  Art.  746,  /(a-l3l)  =  X-Yi. 
Therefore,  a— /3/is  a  root  since  /(a— ^/)  =  0. 

If  the  coefficients  of  /{x)  had  not  been  restricted  to  real  numbers, 
this  theorem  would  not  be  true;  for  S  and  T  in  that  case  might  be 
imaginary,  which  would  prevent  the  application  of  Art.  718. 

756.  If  all  the  coefficients  in  f(x~)  be  cofnmejisurable ,  surd 
roots  of  the  form  a+V  (3  occur  in  pairs. 

This  proposition  may  be  proved  exactly  as  the  above, 
making  use  of  Art.  363  instead  of  Art.  718. 

KXAMPIvKS. 

1.  Form  the  function  of  x  whose  roots  shall  be  2, 

4,  and  —5. 

(x-2)(x-4)(x  +  5)=;t3— ^2_22;*:+40. 

2.  Form  the  function  of  x  whose  roots  shall  be  3, 
.-2,1. 

3.  Form  the  function  of  x  whose  roots  shall  be  3, 

4.  Form  the  function  of  x  whose  roots  shall  be  0, 
3,  |/2,  -1/2. 

5.  Form  the  function  of  x  whose  roots  shall  be  1, 
1,  1,  ~1,  -2. 


RATIONAL   INTEGRAL   FUNCTIONS.  5^5 

Find  all  the  roots  of  the  following  functions,  in  which 
one  or  more  roots  are  given : 

6.     x^-x^+2>x+b',  l-2l/^. 

Another  root  is  1  +  21/"^  and  a  factor  of  f{x)  is  therefore 
j;(;^_1)2+4]  or  x'^-2x^^.  (;r3_;»;24.3^4-5)-=-(^2_2^  +  5)=^  +  l. 
Therefore,  the  roots  are  1-2^,  1  +  2/,  and  —1. 

A  factor  is  x^  ^_i^  and 

Since  ^2  +  4^^.5^[(^  +  2)2  +  l],    the  roots  are    t/-1.  -t/-1, 

8.  x^  +  x^-2bx''+Alx-{-m',  3-t/^. 

g.  ;»;4  +  5;^3__15^2_97^_110;   1  +  V\2. 

10.  x^-x^^-^x^'-^x-lb]  1-21/^,  l/3. 

11.  :r4-5^3-12x2_i3_7.    _l±lzi?. 

757.  I^i  ci'iiy  f{^)^  tJ^^  coefficient  of  the  highest  powe7' 
being  unity,  the  coefficients  of  the  other  powers  a?'e  functions 
of  the  negatives  of  the  roots. 

Since  we  know 

it  is  evident  that  the  values  of /i,  /s'  /yj  ^tc,  can  be 
expressed  in  terms  of  the  roots  by  forming  the  product  of 
the  binomial  factors^  as  in  Art.  91.     Such  product  is  : 

;^"+(— a,-a,—  .  .  .  —a,:)x"-'-\-{aa^  +  aa^+  , .  .  +a„_,a„)^"-^ 
+  ( — aja2a3 — a^aga^ — a^a^o^^ —  •  •  •  — OL,^_^aj^_^a,^x^~^ 

H-  .  .  .  +(  — I)"aia2a3  •  •  •  a„. 

p^,  or  the  fi7'st  parenthesis,  is  the  sum  of  the  negatives  of 
the  roots. 


S86  UNIVERSITY   ALGEBRA. 

/a,  or  the  second  par eiithesis,  is  the  sum  of  the  products 

of  the  negatives  of  the  root  taken  two  at  a  time. 
/>3,  or  the  third  parenthesis,  is  the  sum  of  the  products  of 

the  negatives  of  the  roots  taken  three  at  a  time,  and 

so  on. 
/„,  or  the  nth  parenthesis,  is  the  product  of  the  negatives 

of  all  the  roots. 
Using  the  2  notation,  we  may  express  this  by  writing 

+  •  •  •  +( — I)"aia2a3  .  .  .a„. 
Thus,  in  x^ —  ^x'^A-Wx -^,  —6  is  the  sum  of  the  negatives  of  the 
roots,  -1-11  is  the  sum  of  the  products   of  the  negatives  of  the  roots 
taken  two  at  a  time,  and  —6  is  the  product  of  the  negatives  of  all  the 
roots.     The  roots  being  1,  2,  and  3,  we  verify  by  writing 
-6^-(l-f2+3) 
ll  =  (1.2-t-1.3  +  2.3) 
-6-(-l)(-2)(-3). 
But  in  2x3  — llx2  +  17>r— 6,  ^y^^  sum  of  the  negatives  of  the  roots  is 
not  -11,  for  the  coefficient  of  the  highest  power  of  f{x)  is  not  unity. 
The  sum  of  the  negatives  of  the  roots  is,  in  fact,  —^-. 

758.  If  the  signs  of  all  terms  containing  the  odd  powers, 
or  the  even  powers,  of  x  be  changed,  f{x)  is  t?'ansformed 
into  f{ — x)  or  —f{—x)  respectively ,  the  roots  of  either  of 
the  latter  being  negatives  of  the  roots  of  f(x). 

=  (^— a  J(^— a2)(.^— ag)  •  .  •  {x—a,;).  (1) 

Putting  —X  for  x  throughout  the  equation,  we  shall 

have  f(x)  with  the  signs  of  the  odd  powers  changed, 

equal  to  the  product  of  7i  factors  of  the  form  i—x—a),  or 

I     x''—px''-^  +A-^"~''"~  •  •  •  ^Pn-^^ -^Pn,  if  n  is  even 

=  (— ^--a,)(— ^— a2)(— ^-^3)-  •  '{—X—a,^ 

=  (_l)«(^  +  a,)(^  +  a2)Cr  +  a3)  •  .  .  (.T+aJ  (2) 


RATIONAL   INTEGRAL   FUNCTIONS.  587 

Multiplying  this  through  by  —1,  we  shall  have  f{x)  with 

the  signs  of  the  even  powers  changed,  or 

_/r-  ^  =  l""^"  +/x-^"~'-A-^"~'  +  •  •  •  •\-pn-X'-pn,  if  n  is  even. 

^(_l)«+i(^4-aj(:r  +  a2)(^  +  a3)  •  .  .  (^  +  a,)         (3) 

Now,  by  changing  the  signs  of  the  odd  powers  in/(;r) 
we  must  obtain  what  is  contained  in  the  {  in  (2),  which 
equals  /(—^),  and  from  the  latter  pait  of  (2),  namely: 

(— l)"(^+ai)(^  +  a2)(^  +  a3).  .  •  (^  +  a„), 

it  is  evident  that  the  roots  are  — a^,  —ag,  —o.^,  •  •  •  — ««, 
or  the  negatives  of  the  roots  (1). 

By  changing  the  signs  of  the  even  powers  in  f(x)  we 
must  obtain  what  is  in  the  {  in  (3),  which  equals 
—f{—x),  and  from  the  latter  member  of  the  equation,  it 
appears  that  the  roots  are  the  negatives  of  those  of  (1) 
or  f\x). 

Cha7iging  the  sig7is  of  all  the  terms  of  fix)  does  not 
affect  the  roots.  Changing  all  the  signs  is  multiplying 
by  -1. 

Thus,  the  roots  of  x^  —  Qtx'^  —  llx  —  Q,  being  1,  2,  and  3.  we  know 
that  the  roots  of  either  —x^—Qx^  —  llx  —  Qovx^-\-QiX^-\-llx-\-Q  are 
-1,  -2,  and  -3. 

759.  If  nojie  of  the  coefficients  in  a?iy  fix)  are  fractio7is , 
and  the  coefficient  of  the  highest  power  of  x  is  unity,  the 
f(x)  cannot  have  an  irreducible  fraction  for  a  root. 

Let  such  f{x)  be 

x''  -\-m^x"~^  -\r7n^x"~'^  +  •  •  .  +vi,,_^x-\-7nn, 
in  which  7n^,  7712,  77i^,  etc.,  are  integral.     If  this /(;f) 
can   vanish   when  x  is  an  irreducible  fraction,  let  that 

fraction  be  ^-   Then 


rm- 


on+^^h-Q^i+^2o;;=:2  +  •  •  •  +^n-iQ+m„ 


588  :  UNIVERSITY   ALGEBRA. 

If  the  value  of  this  is  zero,  it  will  be  zero  when  multiplied 
by/5"-i.     But 

^-+;;^la"-l+^^2/^«""^+  •  •  •  +^„-iiS'^"2a+^'^-iw«^0,for 

--  is  an  irreducible  fraction,  since  |  is  ;  and  the  rest    of 

the  function  does  not  contain  a  fraction,  since  none  among 
the  numbers  a,  /?,  m^,  vi2,  m^,  etc.,  are  such;  and  a 
fraction  must  be  combined  with  a  fraction  to  make  zero. 

(X 

Therefore,  it  is  impossible  for  ^  to  be  a  root. 

Note  that  a  f{x)  satisfying  the  conditions  of  the  theorem  need  not 
necessarily  have  integral  roots,  for  some  or  all  of  the  roots  may  be 
incommensurable.  Thus.  x'^—4:X-\-2  has  no  fractional  roots,  but  on 
the  other  hand  the  roots  are  not  whole  numbers,  being,  in  fact,  the 
incommensurable  numbers  2+|/2,  2 — j,/2. 

760.  Any  f{pc)  can  be  transformed  ifito  a  fu7ictio7i  of 
y,  in  which  no  coefficients  shall  be  fractions  and  the  coeffi- 
cierits  of  the  highest  power  of  y  shall  be  unity. 

Suppose /(:r)  in  the  form 

;r"+/>i^"-i  4-/2-^""'  +  •  •  '  +A-2-^'  +A-i-^+A-     (1) 

If  any  of  the  coefficients,  p^,  p^,  /g,  etc.,  are  fractions, 

y 
let  their  common  denominator  be  q.  Put  jj;= -»  then  fix) 

becomes     4;+^^;^!+^^+ .  .  . +A^^ 

^«        q''   1  q''   ■-  q         ^ 

Multiplying  through  by  q",  we  obtain 

in  which  none  of  the  coefficients  are  irreducible  fractions, 

since  each  is  multiplied  by  q,  the  common  denominator. 

Representing  (2)  by  the  symbol  <A(j^),  we  observe  that 

and  that  y^qx. 


RATIONAL  INTEGRAL  FUNCTIONS.      589 

As  an  example,  transform  dQx^  +  '72x^-{-9x-4:  into  a  function 
which  does  not  have  fractional  roots. 

Putting  /{x)  in  the  form  x''  -\- p ^x''- 1 -{-  .  •  .  -|-/„,  we  have 

Taking  x=—,  we  obtain 

Multiplying  by  36^,  we  get 

which  is  in  the  required  form.     A  function  satisfying  the  conditions 

y 

would  have  been  found  by  taking  x:=zw^- 
KXAMPI^KS. 

Transform  the  following  into  functions  which  do  not 
have  fractions  for  roots  : 
I.     4.^3 -15.^2 +  12ji:+4. 

4.     x^-\-2x+Sx'^+x+^. 

5^3    114^2 28-y- 8    . 
X     -i      Q-X          -^jX        2"T- 

6.  3;i;^-40.r3-130jr2-120;i;+27. 

7.  x^—^i-x''--\-x—\. 

8.  The  roots  of  x^-^7x'^-\-16x+12  are  all  negative. 
Find  /(x}  with  numerically  equal  but  positive  roots. 

761.  Any  f{x)  may  be  transformed  into  a  new  functiofi 
whose  roots  shall  differ  by  ayiy  assigned  nnmber  from  the 
roots  of  the  original  function. 

Let  h  be  any  assigned  number  and  y  a  variable  such 
that  y-\-h=x.  Then  we  know  by  Art.  747  that  \i  y-^-h 
be  substituted  for  x  in/(;r),  the  result  is 

/(^)=/0'+/o=/(/o+/'Wj'+/"wf^+...+«oy'  (1) 


590  UNIVERSITY    ALGEBRA. 

Representing /(/2)  by  qn.fQi)  by  qn-i,  etc.,  we  have 

which  latter  expression  let  us  call  <^(jr).  Now,  since 
x~y-\-h,  y  is  alweys  h  less  than  x\  so  also  the  roots  of 
^(jj/)  must  be  h  less  than  those  oi  f{pc). 

To  have  the  roots  <^(jr)  greater  than  those  oi  f{pc),  it 
is  sufficient  to  give  a  negative  value  to  h.  To  have  the 
roots  of  </>(jl/)  less  than  those  of /(x),  it  is  sufficient  to 
assign  a  positive  value  to  h. 

Thus,  toincreasa  the  roots  of  x^  — Gx~-fllj\;— G  by  2, aput  //=— 2 
and  write 

/(-2)  =  (-2)3-6(-2)2  +  ll(-2)-6=-60 
/'(-2)=3(-2)2-12(-2)-|-ll  =  47 
/"(-2)=:6(-2)-12^-24 
/"'(-2)-6. 
Then  the  result  is  /(^)=/(j-2)=  -60  +  47;'— ^-^jj/^  +  fj/^ 

762.  Horner's  Method  of  finding  the  coefficients 
in  <^(jk)  above.  Changing  the  roots  oi  f{pc)  by  an  as- 
signed number  is  such  an  important  transformation  that 
a  shorter  process  than  the  tedious  method  given  above  is 
of  great  value.  We  shall  first  explain  the  new  process 
by  means  of  a  particular  example. 

Take  the  example  given  in  the  last  article.  We  are 
required  to  increase  the  roots  of  x^  —  Qx'^  -i-llx—6  by  2. 
By  the  preceding  article  we  know  that  the  result  of  sub- 
stituting jk — 2ior  xinx^  —  Qx'^'  +  llx—Q  will  be  a  function 
of  the  form 

in  which  we  shall  proceed  to  determine  A-^^  A^^  and  A.^. 
Putting  x=y—2  in  fix),  we  have 

Since  y=^x+2,  we  may  write  this 


RATIONAL   INTEGRAL    FUNCTIONS. 


591 


Dividing  both  sides  of  the  equation  by  x-{-2,  we  have 

jr+2      ^  ^  -^  ^  ;t:+2 

Since   the   dividends   and   divisors  are  equal,  the  re- 
mainders are  equal.     Therefore, 
^3  =  _60. 
-60 


Subtracting  the  equals  —~k  and  -^-^  from  each  side, 


x-^'2 

we  get     ;tr2-8;t:+27==(^+2)2+^^(;r+2)  + ^2- 
Dividing  both  sides  by  ;t:  +  2  again,  we  have 

x-\-2  x+2 

Since  the  dividends  and  divisors  are  respectively  equal, 

the  remainders  are  equal,  and  hence 

^2=47. 

47  A 

Subtracting  — — -.  and  --]\  from  both  sides, 'we  have 

x—\0=x-\-2  +  A^. 
Again  dividing  both  sides  by  x-\-2,  we  have 

^;r+2     ^x+2 
Whence,  ^i  =  ~12. 

We  have  found  the  values  of  ^3,  A,^,  and  A^.  There- 
fore, <^(j/)=jK^  — 12j/2+47j/--60. 

Noticing  that  A^,  A^,  and  A^  are  the  successive  re- 
mainders as  f{x)  is  successively  divided  by  x-^2,  we  can 
perform  the  successive  divisions  by  the  short  method  of 
division  and  for  convenience  arrange  the  works  as  follows: 
1         -6         -fll  -6         (-2 

-2         -fl6         -54 


1 

-8 
-2 

+  27 
+  20 

-60 
II 

1 

-10 
-2 

+  47 
II 
A, 

^a 

1 

-12 

592  UNIVERSITY    ALGEBRA. 

The  above  is  called  Horner's  Method  of  computing  the 
coefficients  of  <j>(y).  We  must  now  consider  the  general 
case.     Suppose 

and  that  we  are  to  find  the  result  of  substituting  y+k 
for  X.     By  Art.  761,  we  have 
a^x"-}-  -  .  .  +a,,_2x'^  +a,,_^x+a,, 

=A,r+  .  .  .  +A,,_,y^+A,,_,y+A,^,  (1) 
in  which  we  are  to  explain  a  short  way  of  finding  the 
values  of  ^0'  ^i>  ^^c- 

SincQ  x=jy-j-k,  y=x—k,  and  making  this  latter  sub- 
stitution in  right  member  of  (1),  we  have 

=Ao(x^hr+  •  •  ~  +A^,_,(x^/iy+A^,_,Cx-/i)  +  A,,  (2) 

Divide  both  sides  of  (2)  by  {x — k)  and  call  the  quotient 

in  the  left  member  Q-^  and  the  remainder  J^^ .  We  then  have 

+^„_2(-r~/0+^«_i+^|-       (3) 

Since  the  dividends  and  divisors  are  equal,  the  re- 
mainders are  equal,  and  we  have 

That  is,  the  last  undetermined  coefficient  in  <^(jO  equals 

the  remainder  when  /(x)  is  divided  by  x—/i, 

R  A  ' 

Subtracting ^  and  — —  from  the  members  of  (3), 

we  have 

^1=^0(^+^0""'+  •  •  •  4-^„-2(^-/0+^.-i.     (4) 
Again  dividing  by  x—h  and  calling  the  quotient  in  the 
left  member  Q^  and  the  remainder  Rc^^,  we  have 


RATIONAL    INTEGRAL    FUNCTIONS.  593 

Whence  it  is  seen  that  ^4„_i ,  the  next  to  the  last  undeter- 
mined coefficient  in  <A(jk)  equals  the  remainder  when  Q^ 
is  divided  by  x—k;  and  so  on.     Hence: 

T/ie  coefficients  in  <^(j|/),  beginyiing  at  the  last,  are  the 
successive  re7nai7iders  as  f{x)  is  co?itinuously  divided  by  x — h. 

As  another  application,  suppose  it  is  required  to  reduce  the  roots 
of  :if^  — 5x2— 5:)C-f-25jby  2.  We  divide  successively  by  x— 2  by  syn- 
thetic division  as  follows: 

1     —5     —5     +25  (2  Abreviated  sometimes  thus : 

-6     -22  1     -5     ~5     -25  (2 


1     -3 
2 

-11 
—  2 

3 

1! 

1     -1 
2 

-13 

II 
A. 

^3 

I  i     1 

II  II 
Aq     A^ 

1     -3  -11  1 

3 

1     -1|-13 

1    1    1 
1 

Result  :  ;/3+;/8-13jj/-!-3. 
EXAMPI.KS. 

Change  the  roots  ot  the  following  expressions  by 
Horner's  Method  : 

1.  ^4  +  4;r3— 2;t:2  — 12  +  9,  roots  to  be  3  greater. 

2.  x^—9x'^-{-2Sx—15,  roots  to  be  1  less. 

3.  x^—Qx—lS,  roots  to  be  3  less. 

4.  x^  +  3;t:2  — 10;r— 5,  roots  to  be  2  greater. 

5.  x^—Sx^-^x'^+4:,  roots  to  be  2  less. 

6.  6x^+2Sx-^-72x-{-S6,  roots  to  be  4  less. 

763.  Removal  of  Terms.  If /?  be  selected  properly, 
/(y+h)  can  be  made  to  lack  any  specified  term,  except 
the  first.     For  take 

Then  putting  y  +  h  for  x 

/(y+h)  =  (y+hr+P,(y-\-hy-^+P,(y+hr-'+  .  .  .  4-^ 
=y'+(Pi-^nh')y-^-{-(^p,-\-p,nh-{-'^^^h^'jy»-^ 


594  UNIVERSITY   ALGEBRA. 

by  expanding  by  binomial  theorem,  as  in  Art.  747,  except 
.  that  we  have  here  arranged  the  result  according  to  the 
descending  powers  of  y. 

Now  the  next  highest  power  of  j/  will  not  occur  in  the 
transformed  function  if  the  coefficient  /^  +7^^=0;  that  is, 

if  A= — ^^-     So  that  zY  h  be  taken  equal  to  — -—^      the 

next  highest  pozver  of  the  variable  will  be  wanting^  after 
applying  Horner*  s  Method. 

Likewise,  the  second  highest  power  will  not  occur  in 
the  result  if  the  coefficient 

If  this  quadratic  be  solved  for  h,  two  values  of  h  will, 
in  general,  be  found.  If  either  value  of  h  be  used  in 
Horner's  Method,  the  transformed  function  will  lack  the 
second  highest  power  of  the  variable,  and  so  on. 

As  an  example,  remove  the  second  term  of  ^i:^— 6^^4-5.  Here 
«=3  and  /^=:-6.  so  h=%.     Then 

1         -6         +0         +5       (3 


2 

-8 

-16 

1 

-4 

-8 

-11 

2 

-4 

1 

-2 
2 

-12 

1  0 

Result:  x^ -Vlx-\\,ox  y^  —  Vly  —  W.  It  is  customary  to  use  x  as 
the  variable  in  such  transformations,  although  of  course  it  is  not  the 
same  x  as  the  one  in  the  original  function. 

KXAMPLKS. 

Remove  the  second  term  in  each  of  the  following: 

1.  ;r3+6;i;2_i3^_p42. 

2.  ;i;3-9;i;2  +  23;i;— 15. 

3.  x'^  —  ^Sx^+A^x^—ex+ll, 

4.  x^-Sbx''  +  (Sb''-a'')x-b(ib'''-a''), 


RATIONAL  INTEGRAL  FUNCTIONS.      $95 

5.  ;t:3  +  9ji;2+27;f+27. 

6.  x^'-ex^  i-Ux^-lSx'^-i-Ux'^-Qx^l. 

7.  Sx'^—4x^  —  Ux'^—4:X+S. 

8.  Transform  ax^-{-dx^-{-cx+d,  in  which  a,  b,  c,  d 
are  whole  numbers,  into  a  function  of  the  form  2^-\-l2+ni, 
in  which  /  and  m  are  whole  numbers. 

PROPERTIKS  AND  CONSTITUTION  OF  DKRIVATIVKS  OF/(;t). 

764.     Maxima  and  Minima  Values.    A  Maximum 

value  of /(:f)  is  a  value  that  is  greater  than  the  immedi- 
ately preceding  and  succeeding  values  of /(jtr).  A  Min- 
imum value  of  f{x)  is  a  value  that  is  less  than  the 
immediately  preceding  and  succeeding  values  of /(;i;). 

Any  f{x)  may  have  several  maxima  and  several 
minima  values.  Thus,  if  the  variation  iij  value  of /(jr) 
as  X  is  changed  be  represented  by  the  varying  distance 
of  a  point  moving  in  the  curved  line  from  the  line  OX^ 
then  in  figure  20, 


Fig.  20. 

the  points  A,  C,  and  E  correspond  to  the  minima  values 
of /(:r)  and  B  and  D  correspond  to  maxima  values. 

In  symbols,  if  f(a)  be  a  maximum  value  of  f(x)  and 
h  be  a  small  number,  then  we  must  have 

f(a)>f(^a-\-h)  and  also  /(^)>/(«-/^). 

li  f(b)  be  a  minimum  value  of /(;i-),  we  must  have 

/(^)</(^^  +  /0  and  also /(^)</(<5-/^). 


596  UNIVERSITY    ALGEBRA. 

765.  If  ci  is  any  root  of  f\x)^  then  if  f"(x)  is  positive ^ 
f{a)  is  a  minimum  value  of  f{x)  and  if  f"{a),  is  a  nega- 
tive^ J  ^a)  is  a  maximum  value  of  f(^x) 

From  [1],  Art.  747,  we  have 

/(^+/^)~/(a)=/'(a)/^+/''W^+ ...  (1) 

Changing  the  sign  of  h,  we  get 

f^a-h-)-f{a)=-f{a)h+f"(a)~ (2) 

But  since  f(a)=Oy  by  supposition,  these  become 

f(a+k)-f(a)=/"ia-)^^+  ...  (3) 

/(.a-h)-f(a)=^f"{a)^ (4) 

Now,  h  may  be  taken  so  small  that  the  first  terms  in 
the  left  members  will  be  numerically  greater  than  the 
sum  of  all  the  terms  that  follow,  by  Art.  749.  Therefore, 
if  y*"(«)  is  positive,  we  have,  from  (3)  and  (4), 

f{a)<f{a-^K)  ^nd.  f{a)<f{a-1i). 

Therefore, /(a)  is,  by  definition,  a  minimum  value  of/(;r). 
But  \i  f"{a)  is  negative,  then  we  derive  from  (3)  and  (4) 

/W>/(^  +  /0  and  f{a)>f{a-^Ji), 

in  which  case  f{a)  is  plainly  a  maximum  value  oi  f{pc), 

(1).     Find  the  max.  and  min.  values  of  ^:X^ — 15^^^  ^12^— 2. 

/"(x)=:24;*:-30 
The  roots  of  /'  {x)  are  \  and  2. 

/"(i)=-18  .-.  /(i)=|=max.  value  of /(^). 
/"(2)=-f  18.-.  /(2)=-3:=min.  value  of /(^). 

(2).  Find  the  values  of  x  for  which  3^^5  —  125^''  +  21 60.r  is  a  max. 
or  min, 

/'(x)==15x4-475;c24-2160 
/"(;^)=G0;tr3-950^-. 


RATIONAL   INTEGRAL   FUNCTIONS.  597 

Roots  of  f'{x)  are  -4,  -3,  3,  and  4.  Use  6^3  _95;^  for  f"{x) 
and  X*  - 35^2 _|_  144  for /'(x). 

/"(_4)=-4  .-.  /(-4)  =  max. 
/"(_3)=rl23  .♦.  /(-3)=min. 
/"(3)=-123  .-.  /(3)=:  max. 
/"(4)=  4  .-.  /(4)=  min. 
Therefore,  —4  and  3  are  the  values  of  x  that  render  f{x)  a  max., 
and  —  3  and  4  are  the  values  of  x  that  render  f{x)  a  min. 

766.  ^f  f{p)  ^^  ^  maximum  or  a  minimum  of  f{x)^ 
f\a)  va7iishes. 

This  follows  at  once  from  (1)  and  (2)  above,  for  when 
k  is  very  small  the  left  members  of  (1)  and  (2)  cannot 
have  like  signs  unless /'(^)=0. 

767.  Between  two  consecutive  real  roots  a  and  b  of  f{x), 
there  is  at  least  07ie  real  root  of  f\x). 

Since  a  and  b  are  consecutive  roots,  fix)  is  not  again 
zero,  as  x  changes  from  a  to  b.  Therefore,  as  x  varies 
from  a  to  b,  fix)  must  either  at  first  increase  and  then 
decrease,  in  which  case  fix)  has  a  maximum  value;  or 
fix)  must  at  first  decrease  and  afterwards  increase,  in 
which  case  fix)  has  a  minimum  value.  In  either  case, 
the  value  of  x  which  renders  fix)  a  maximum  or  a  min- 
imum, is  a  root  of /'(•^). 

768.  Constitution  of /'(aj),  in  terms  of  factors  of /(;r). 
We  know 

fix)  =  ix—a^ix—a^iX'-a^.  •  •  (:r— a„). 

We  seek  an  expression  for  fix)  in  terms  of  these  same 
factors  x—a.^,  x—o.^^  etc. 

Put  h-\-x  for  X,  we  then  have 
/(>^  +  ^)  =  [/^  +  (^-ai)][/^+(ji;-a2)]+.  •  -C/^  +  C^-a,,)] 


598  UNIVERSITY   ALGEBRA. 

in  which,  by  Art.  91,  we  know 

^i  =  Gr— ai)  +  (;i;— a2)  +  (;r— a3)-f  ■  •  .+(^~aj 

.  .  4-  •  •  •  +(.^-^«-i)(^-—a,,) 
+  .  .  .  +(^— ai)(;r— ag)  •  •  •  (x—a,_^) 

But  by  [1],  Art.  747,  we  have 

whence  we  have,  equating  coefficients  of  powers  ol  //  in 
the  two  expressions  for  /(/i+x), 

fCx')  =  (_X-a^X^-^s)"-(.^-<^n')  +  (x-a^)(x-a.^).,.(^-^n) 
+  •  •  •  +(^--tti)(;r--a2.)  .  .  .  Cx—a,,_^)  [4] 

etc.  etc. 

The  expression  for  /'(^)  may  be  written  in  the  useful 
form 

X — a^      x—a^  ^—oLn 

Thus,  if  /(x)^x^-Qx^  +  nx-G=(x-l){x-2){x-^) 
then  /' {x)^{x-2)lx-3)-\-{x-l){x-3)  +  {x-l){x-2) 
_/(^)    ,    /(^)    ,    /(^)_ 

769.  Equal  Roots.  // /(x)  has  r  roots  equal  to  ti.  f'{x) 
has  r—1  roots  equal  to  a. 

This  can  be  seen  from  the  form  of /'(-''^)  found  above: 
/'(^)  =  (^-a2)(:f-a3)...(;j;-aj4-(^-ai)(^-a3)...(;r-a„) 

+  •  '  •  +(-r— ai)(;r— a2)  •  •  •  (^^~a«._  J. 
For  if  ai=a2,  then  f'(x)  is  seen  to  be  divisible  by  x—a^ 
and  hencea^  is  a  root  of/' (;t:).     l{  a^—a^—a.^^  then  f\x) 


RATIONAL   INTEGRAL   FUNCTIONS.  599 

is  seen  to  be  divisible  by  (x—a^y-,  and  so  on.  Thus,  if 
/(x)  is  divisible  by  (^— a)'',  then  /'(x)  is  divisible  by 

Note  that  if /(x)  has  equal  roots, /(:r)  and  /'(^)  have 
a  common  divisor. 

(1).     Thus,  if /(x)=;^3_5^2_^7^_3=(^_l(^_l)(^_3) 
then  /'{x)={x-l){x-:\)  +  {x-l){x-3)i-{x-l){x-l), 

and  f'{x)  is  seen  to  be  divisible  by  x  —  1. 

(2).    The  expression  x^  -  Ix'^-hlQx  -  12  has  equal  roots;  find  them. 

/'{x):^3x^-Ux-^lQ. 
The  H.  C.  F.  of  /(x)  and  /' (x)  must  now  be  found. 

3x^-2lx^  +  ASx-m     I     3x^-Ux-^lQ 
^x^-Ux^-hlGx  I     x  +  7 

-  lx'^-{-'S2x-  36 
21^2 -96;r+ 108 
2U=^-98x  +  112 

2x—     4,  etc. 

whence  X— 2  is  a  common  factor.  Therefore,/(x)=(;t:  — 2)(^— 2)(jr:— 3), 
in  which  the  factor  {x — 3)  is  found  by  dividing  /{x)  by  {x-2)^. 

BINOMIAIv  COEFFICIENTS. 

770.     In  the  theory  of  the  rational  integral  function 
it  is  often  advantageous  to  represent  /(x)  as  follows: 

7l(fl  —  1^ 

a^x''+7ia^x''~'^-\ — z — ^a<y^x''~'^-\-  -  .  . 

,  nGi—V)  _  , 

H :j — K-^n-2^  +na,,_^x+a^, 

in  which  the  successive  coefficients  in  (x+a)''  are  given 
to  the  terms  in  addition  to  the  coefficients  a^,  a^,  etc. 
It  is  plain  that  any  /(x)  can  be  expressed  in  this  form, 
and  when  so  expressed  it  is  said  to  have  Binomial  Co- 
efficients. Note  that  /(^),  written  in  the  above  form, 
has  the  same  appearance  as  the  expansion  of  (x+a)", 
except  that  a  occurs  with  subscripts  instead  ot  exponents. 


600  UNIVERSITY   ALGEBRA. 

771.  When  a/(^)  of  the  nth  degree  is  expressed  with 
binomial  coefficients  we  shall  represent  it  by  the  symbol  U„. 
Thus, 

n(7i—l) 
Diminishing  n  by  unity,  we  get.  in  succession, 

U'^  =  aQX^  +Sa^x'^  +Sa2X+a^. 
Ui  =  aQX+aj, 

772.  Derivatives  of  Un.  We  may  express  the  first 
derivative  of  17^,  as  written  above,  as  follows: 

n\aQx''-'^  +  Cn—l)a^x"-^  +  •  •  •  +{7t—l)a,,_2^+a„_^']. 
That  is,  the  first  derivative  of  Z7„  is  nUn-\.  Thus,  the 
first  derivative  of  U^  is  found  by  multiplying  by  the  sub- 
script of  Uj^  and  decreasing  the  subscript  by  unity.  This 
is  the  same  process  as  explained  in  Art.  747,  except  that 
it  is  here  applied  to  subscripts  instead  of  exponents.  Thus, 

we  have 

derivative  of  17,,     =  n  17,,- 1 . 
derivative  of  ^„_i  =  (n—l)  U,,^^  J 

derivative  of  Z/g    =3  U^ ; 
derivative  of  ^2    =26^^,  etc. 

773.  The  f(x+h)  with  Binomial  Coefficients.  We 
know  f(ix+h)  may  be  expressed  as  follows: 


RATIONAL    INTEGRAL    FUNCTIONS.  6oi 

Representing /(;r)  by  6^„,  then  since /i(^)  =  ;z6^„_i,  etc., 
we  have 

Interchanging  x  and  h  and  representing   the    result  of 
substituting  h  for  x  in  U^  by  A,,,  we  shall  have 

In  this  expression,  remember  that 

^^0  =  ^0,  ^i=^o^+ai,  y^2  =  ^o'^^^+2«i'^  +  ^2>  etc. 


CHAPTER  XXXIII. 

SPKCIAL  EQUATIONS. 

774.  The  treatment  of  rational  integral  equations  is 
based  upon  the  properties  oi  f{x),  as  considered  in  the 
last  chapter.  The  present  chapter  is  devoted  to  a  few 
important  special  classes  of  rational  integral  equations, 
namely:  reciprocal  equations,  binomial  equations,  cubic 
equations,  and  biquadratic  equations. 

RKCIPROCAIy   KQUATIONS. 

775.  If  an  equation  is  unaltered  by  changing  x  into 
->  it  is  called  a  Reciprocal  Equation.  Thus,  x^+5x^ 
--10;i;2  +  5;i;+l=0  is  a  reciprocal  equation. 

776.  Classes  of  Reciprocal  Equations.  Consider 
the  equation 

x"+p^x"-^+p^x''-^+  .  .  .  +A-2-^''+A-i^+/«=0. 

Putting  —  for  x,  and  clearing  the  result  of  fractions, 
or  x"+^x"-'-h^'x"-^+  .  .  .  +l^x^+llx+^=0. 

Pn  Pn  Pn  pn  Pn 

If  the  original  equation  is  unaltered  by  this  substitution, 
we  must  have 

pn  pn  pn  Pn  pn 

From  the  last  equation  we  have  /„==fcl,  and  thus  we 
have  two  classes  of  reciprocal  equations  : 


SPECIAL  EQUATIONS.  603 

I.  If  /!,=  !,  we  have 

P\=Pn-\,  Pl=Pn-1y  •    •    •  ,  Pz=pn-Z,  etC. 

That  is,  the  coefficients  of  terms  equidistant  from  the  ends 
are  equal. 

II.  If  jz^„=—- 1,  we  have 

Pl^^—Pn-I,  P'i  =  —pn-2^  Pz='-Pn-Z^^^^' 

That  is,  the  coefficients  of  terms  equidistant  from  the  ends 
are  numerically  equal  but  of  opposite  signs. 

If  the  equation  happens  to  be  of  an  even  degree,  say 
2r,  then  pr=  —pr,  or  j2^^=0.  Thus,  if  an  equation  of  this 
class  is  of  an  even  degree,  the  middle  term  is  wanting. 

777.  Roots  Reciprocals.  Since  a  reciprocal  equa- 
tion is  unaltered  by  changing  x  to  — »  it  follows  that  if  a 

is  a  root  of  a  reciprocal  equation,  —  must  also  be  a  root. 

a  11 

Thus,  roots  enter  in  pairs  of  the  form  a,,  — ;  a^,  — ,  etc. 

If  the  equation  be  of  an  odd  degree  it  must  have  a  root 
which  is  its  own  reciprocal,  and  as  +1  and  —1  are  the 
only  numbers  of  this  kind,  one  of  these  must  be  a  root. 

778.  If  a  pair  of  reciprocal  roots,  or  any  root  that  is 
its  own  reciprocal,  be  removed  from  the  equation  by 
dividing  its  left  member  by  the  proper  expression,  then 
the  resulting  equation  still  has  its  roots  in  reciprocal 
pairs,  and  therefore  is  a  reciprocal  equation. 

779.  Reduction  to  a  Single  Class.  A  reciprocal 
equation  of  class  I,  say 

or         Gr"+l)+/>i^(-^-"+l)+/>2-^'(-^"  +  l)+  •  •  •  =0, 

of  an  oda  degree  has  a  root  —1,  because  its  left  member 


604  UNIVERSITY   ALGEBRA. 

is  divisible  by  :r+l.     If  ^(;r)    be   the    quotient,    then 
<^(;j;)  =  0  is  a  reciprocal  equation  (Art.  778)  of  an  even 
degree,  having  its  last  term  positive. 
A  reciprocal  equation  of  class  II,  say 

or         (:r«-l)+/i^C;i:«~l)+/2-^'(^"-l)+  •  •  •  =0, 
of  an  odd  degree  has  a  root  +1,  because  the  left  member 
is    divisible    by  .:r— 1.     If  ^{x)   be   the   quotient,   then 
<^(x)=0  is  a  reciprocal  equation  (Art.  778)  of  an  even 
degree,  with  its  last  term  positive. 

A  reciprocal  equation  of  class  II  of  an  even  degree  has 
a  root  4-1  and  a  root  —1,  because  its  left  member  is 
divisibleby  ^2— 1.  \i<^{x)  be  the  quotient,  then  (^(:r)=0 
is  a  reciprocal  equation  (Art.  778)  of  an  even  degree,  with 
its  last  term  positive. 

Thus,  we  have  shown  that  any  reciprocal  equation  may 
be  reduced  io  one  of  even  degree^  with  its  last  term  positive. 
This  may  be  called  the  Standard  Form  of  the  reciprocal 
equation. 

780.  Any  reciprocal  equation  in  the  standard  for7n  may 
be  reduced  to  an  ordinary  equation  whose  degree  is  half  that 
of  the  standard  form. 

Let  the  equation  in  the  standard  form  be 

^2^+i^i^'"-'+  •  •  •  +A^''+  •  •  •  +/i^-f  1=0. 
Dividing  by  x'',  and  grouping  terms  equidistant  from  the 
ends,  we  have 


Now,  x*-\-—^  may  be  expressed  as  follows  : 


SPECIAL  EQUATIONS.  605 

Giving  p  the  values.2,  3,  4,   •  .  • ,  in  succession  and  rep- 

1  1 

resenting  x-\ —  hy  u  and  x^-\ — ^  by  V^,  we  get 

F2  =  7^2_2         or        :r2+-4  =  ^^'-2. 
2  x^ 


Vr=u  Vr-i  —  Vr-2  y  ^nd  SO  on. 

Substituting  these  values  above,  we  get  an  equation 
of  the  rth  degree  in  7c.  From  the  values  of  u  the  values 
of  X  may  be  found  by  solving 

X-] =71. 

X 

(1).     Solve  x4- 10x3  4- 26x2 -10;c4- 1=0. 
Dividing  by  x"^,  we  get 

(.^.+l^^_lo(x+i]  +  26=0. 

Substituting  the  values  of  Fg  and  V^  in  terms  of  u,  we  get 

«2_  10^4. 24=0. 
Whence,  «=4  or  6. 

Then,  x+— =4  or  6, 

X 

or  jr3 - 4^^+1=0  and  x2_  (3;^; 4. 1—0 

Therefore,  x=z2±\/T,     3±2v/2: 

(2).     Solve  2;»:5-15;»r4+37x3-37;c2  +  15-r-2=0. 
Dividing  by  ^  —  1, 

2     -15     +37     -37     4-15     -2    (1 

2     -13        24     -13        2 
2     -13        24     -13  2        0 

we  get  2^4_i3;^3_|.24x;2-13;«:+2=0. 

Dividing  by  x^,  we  get 


2(.»+l)-13(.+j)  +  24=0 


Substituting  the  values  of  V^  and  F"^  in  terms  of  u,  we  get 

2z<2-13z/+20=0. 
Whence,  «=4  or  f . 


6o6  UNIVERSITY    ALGEBRA. 

Then,  from  x-\-~=4:Ot% 

X  ^        ' 

we  get  x=2,     i      2±\/W. 

Therefore,  in  the  original  equation 

x=\,  2,   1,   2±\/3: 

(3).     Resolve  x"^  +  1  into  quadratic  factors. 

We  know  x'^  +  l—O  is  a  reciprocal  equation.     Put  it  in  the  form 


or 

w2-2=:0. 

Therefore, 

u=±Vr. 

That  is 

X                                                 X 

or 

^2-v/^.r+l=r0and^2  +  \/2'x4-l=0. 

Hence, 

^4  +  1^(X2  -  v/  ^^--M)(;t:2  ^.  ^2  ;c  +  l). 

EXAMPLES. 

Solve  the  following  equations : 
I.     2;^4__5^3^6^-2_5^_^2=0. 

3.  :^;4+5^3  +  2;t:2+5;r+l=0. 

4.  x^  +4:ax^+2x^  +Aax^  +  1=0. 

6.  2;i:<5+Ji:'^  — 13;i;4  +  13;i;'^— ;tr— 2=0. 

7.  Two  roots  of 
x^^llx''+iSx^-79x^+79x'^—i7x+'i:0=0sire  2  and  5 
Find  all  the  other  roots. 

8.  Show  that  _ 

x^  +  l  =  (x'^  +  l)(x'^  +  VSx+lXx^  --l/Sx+l). 

BINOMIAI,  EQUATIONS. 

781.  An  equation  of  the  form  ;r'*=fc^=0,  in  which 
a^O,  is  called  a  Binomial  Equation.  This  equation 
has  n  roots  by  Art.  752. 


SPECIAL  EQUATIONS.  607 

782.  Roots  all  Different.  A  binomial  equation 
cannot  have  equal  roots.  For,  since  f(^x)=^x"^a,  we 
\i2N^  f  {x)  —  nx''-'^ ,  But  x^'^^a  andnx"~^  have  no  com- 
mon divisor.  Therefore,  of  the  n  roots  of  a  binomial 
equation,  no  two  can  be  alike. 

783.  Roots  of  a  Complex  Number.  The  demon- 
strations in  Arts.  751  and  752  hold  whether  the  coefficients 
ofyC^)  be  complex  numbers  or  not.  Therefore,  if  a  be 
a  complex  number,  the  equation  x''—a=0  or  x"=ahas  n 
roots,  which  we  have  just  seen  are  all  different.  But 
every  root  of  the  equation  x''—a  must  be  a  value  of  i^' a. 

Therefore,  every  coinplex  member  has  n  differejit  nth  roots 
and  no  more. 

Since  a  positive  real  number  is  a  complex  number  ot 
amplitude  0,  and  every  negative  real  number  is  a  complex 
number  of  amplitude  180°,  it  follows  that  every  positive 
or  negative  real  nu^nber  has  n  nth  roots  a7id  no  m^ore. 

784.  Suppose  the  complex  numbers  to  be  represented 
by  r(cosO + /  sin^) .    "Then  the  following  n  expressions  are 

The  n  nth  Roots  of  the  Complex  Number  a, 

V  6>         .    ^\ 

?"  cos--f  ^sm-  (1) 

\      n  n)  ^  ^ 

V  27r+^   ,    .    .    27r-f  (9\ 

r'\  cos h  t  vSm )  (2) 

\  71  n    I  ^  ^ 

r "(  cos h  I  sm j  (3) 

r«(cos h^sm )  (s-\-Y) 

\  n  71     J  ^ 

r"[ Qos- h  z  sm^ --]  (71) 

\  n  n  )  ^  ^ 


6o8  UNIVERSITY   ALGEBRA. 

For,  take  the  nth  power  of  any  of  these  7z  expressions,  as 
the(^+l)st.     Then 

r«(cos f-2sm — J 

/        s27r  +  e   ,    .    .    s2'7r  +  e\" 

=  r[  cos h  z  sm ) 

\  71  71       I 


„(, 


cos-^ — --\-i  sm-^ — -j 


by  De  Moivre's  theorem, 
=  r[cos(527r+(9)  +  / sin(^27r+l9)] 
=  r(cos  0-\-i  sin  &) 
whatever  integral  value  ^  may  have. 

Since  the  7^th  power  of  any  one  of  the  n  expressions 
written  above  is  r(cos  ^+/sin  ^),  and  since  no  two  of 
these  expressions  are  alike,  therefore  these  7i  expressions 
are  the  7i  Tith  roots  of  r(cos  ^+/sin  0)  or  a,  which  in 

Art.  783  were  shown  to  exist. 
1 
Note. — Remember  that  r"  in  the  above  stands  for  the  positive  or 
arithmetical  value  of  the  n'Oa.  root  of  r. 

785.  In  the  71  expressions  written  in  the  last  article, 
put  ^=0  and  we  shall  have 

The  n  nth  Roots  of  a  Positive  Number,  r. 

^"  ,         .  (1) 

1/        27r  ,    .    .    27r\ 

r'\  cos h  I  sm — )  (2) 

\       n  7t  J  ^  ^ 

1/        47r  ,    .    .    47r\  ^^^ 

r'i  cos 1- 1  sin — )  (3) 

r«  cos Vi  sm —  U+1) 

\        ^  n  )  ^        ^ 

1/       (;^-l)27r  ,   .    .    (>^~l)27r\ 
r «( cos-^ h  I  sm- —  I  (ft) 

\  71  71  J  ^ 


SPECIAL  EQUATIONS.  609 

The  first  of  these  is  real.     If  n  is  an  even  number,  say 

2^,  then  the  (^+l)st  expression  reduces  to 
\_  ^  1 

r''(cos  TT+z  sin  7r)  or  — r" 

That  is,  if  n  is  even  there  are  two  real  roots  of  opposite 

signs,  but  all  the  remaining  roots  are  imaginary. 

786.  In  the  n  expressions  of  Art.  784,  put  ^=7r,  and 
we  shall  have 

The  n  nth  Roots  of  a  Negative  Number  —r, 

r"(  cos-  +  /  sin- )  (1) 

\      n  n)  ^  ^ 

r"\  cos V I  sm — )  (2) 

r"  cos Vt  sm —  (3) 

\       71  n  I  ^  ^ 

r«  cos-^^ ^ — \-t  sm- —  (^+1) 

V  71  71  J  ^ 

1/      (2;z~l)7r  ,   .    .   (2n-V)iT\ 
r'icos- — [-1  sm- )  (n)^ 

\  71  71         J  ^    ^ 

If  71  is  an  even  number,  all  of  these  expressions  are 
imaginary.     If  71  is  an  odd  number,  say  2^+1,  then  the 

(5+l)st  expression  reduces  to  —r"  and  all  the  rest  are 
imaginary.     That  is,  if  7i  is  odd  there  is  one  real  root  of 

— r;  namely:  the  negative  number,  — r«. 

787.  If  in  the  7i  expressions  in  Art.  785  we  put  r=l, 
we  shall  obtain  the  7i  ?ith.  roots  of  + 1.  The  first  imagin- 
ary 7zth.  root  of  + 1  is  called  a  Primitive  nth  Root  of  +  J, 
and  is  represented  by  the  symbol  w;  that  is, 

27r  ,    .     .    27r 

COS \-i  sm — =0).  • 

71  71 


6lO  UNIVERSITY   ALGEBRA. 

Then,  by  DeMoivre's  theorem, 

47r  47r 

cos ^z  sm — =0)2. 

'  n  n 

cos Yz  sm — =0)3. 

and  so  on  for  the  other  ;?.th  roots  of  +1.     The  n  power 
of  the  primitive  root  gives  us 

cos Vi  sin — )  =cos  27r+/sin  27r=  +  l, 

\       n  7zJ  ' 

the  first  ni\\  root  of+1.  Therefore,  the  following  is  true: 

If  0)  represents  a  priyiiitive  7ith  root  of  +1,  then  all  the 

nth  roots  of  -\-\  are  represented  by  the  expressions  o),  o)^, 

O)^     .    .    .,    0)". 

Since  the  ;^th  roots  of  +;    differ   from  those  of  +1 

1 
merely  by  the  presence  of  the  power  of  the  modulus  r'\ 

we  may  say: 

If  r  is  a  positive  number  arid  w  a  primitive  nth  root  of 

-f  1,  then  the  n  nth  roots  of  r  are  represented  by  the  expres- 

11  1 

sions  r"o),  r"o)2,   •  •  •  r"ni^. 

788.  If,  in  the  n  expressions  in  Art.  786,  we  put 
r=^l,  we  shall  obtain  the  n  n\h  roots  of  —1.  The  first 
imaginary  ;^th  root  of  —1  is  called  a  Primitive  nth  root 
of  — 1,  and  is  represented  by  o>';  that  is, 

TT  .       .       TT  , 

COS— +  2  sm— =0)  . 

71  71 

Then  it  is  easy  to  see,  by  De  Moivre's  theorem,  that 
the  following  is  true: 

If  o)'  represent  a  primitive  nth  root  of  — 1,  then  all  the 
nth  roots  of — 1  are  7'eprese7ited  by  the  expressio7is  o)',  w'^^ 

0)'3,      .    .    .,    0)'". 

Since  the  ?eth  roots  of  —r  differ   from  those  ot    —1 

merely  by  the  presence  of  the  power  of  the  modulus  r**, 
we  ma}^  say: 


SPECIAL  EQUATIONS.  6 1 1 

If  —r  is  a  negative  number  and  <i>'  a  primitive  7ith  root 
of  —1,   then  the  n  7ith  roots  of  — r  are  represented  by  the 

expressions  r"  oj',  ^«  co'^^    .  .  .^  ;^«  o)''*. 

These  same  roots  may,  however,  be  obtained  by  multi- 
plying any  one  of  them  by  the  7i  different  72th  roots  of  +1. 
Therefore,  we  may  say  the  n  nth  roots  of  any  mmiber,  pos- 
itive or  negative^  may  be  obtained  by  i7iidtiplying  any 
particular  07ie  of  the  nth  roots  by  the  different  nth  roots 
of  +1.      That  is,  by  w,  w^,  w^,  -  .  •  w"". 

789.  As  applied  to  the  binomial  equatio7i  x"  +  a=Oy 
we  have  shown: 

I.  In  case  a  is  imaginary,  there  are  n  different  im- 
aginary roots. 

II.  In  case  a  is  positive,  (1)  there  is  one  negative  real 
root  and  (71  — 1)  different  imaginary  roots  if  n  is  odd,  and 
(2)  there  are  n  different  imaginary  roots  if  n  is  even. 

III.  In  case  a  is  negative,  (1)  there  is  one  positive 
real  root  and  (71  — 1}  imaginary  roots  if  n  is  odd,  and  (2) 
there  is  one  positive  and  one  negative  real  root  and  (ji—2) 
imaginary  roots  if  n  is  even. 

790.  If  r  and  t  be  prime  to  each  other,  the  equations 
x'' — \=^^ a7id  x^ — 1  =  0  have  no  com7non  root  except  -f  1. 

The  amplitudes  of  the  roots  of ;»;''— 1  are  seen,  from  Art. 
787,  to  be 

^     2ir     47r  (r— l)27r 

(J    — ,  — ,    .  .  . .. 

r      r  r 

Likewise,  the  roots  of  ;t:^— 1=0  are 

^     27r     Aw  {t—iyiir 

The  common  amplitude  0  indicates  the  common  root 
-f  1,  but  no  fraction  in  the  first  series  can  equal  a  fraction 
in  the  second  series,  since  r  and  /  are  prime  to  each  other. 


6l2  UNIVERSITY    ALGEBRA. 

Since  the  amplitudes  of  the  roots  of  ;t:''  —  l=0,  except  0, 
differ  from  the  amplitudes  of  the  roots  of  x'--l=0,  the 
equations  have  no  roots  in  common,  except  +1. 

791.  If,  in  the  equation  jr"— -1  =  0,  7i  is  the  product  of 
two  numbers  s  and  /,  which  are  prime  to  each  other,  theji 
the  roots  of  ;r"— 1  =  0  can  be  found  from  the  roots  oj 
ar^-l  =  Oand;r^-l=0. 

In  fact  if  a  be  a  primitive  root  of  ^'■—1=0  and  P  a 
primitive  root  of  :r'— 1  =  0,  then  the  7t  roots  of  .r"— 1  =  0 
will  be  the  n  ter^is  in  the  product 

(a4-a2+a3+.  .  .  -^aOC/^  +  i^^ +^3  _^  .  .  .  ^^.)         (^y 

For,  let  a^/3^  be  any  term  in  this  product.  Now  (a^)'"=l, 
since  a^  is  a  root  of  ;tr^— 1=0;  also  (yS^)^=l,  since  P^  is  a 
root  of  ;r'— 1=0.  Since  (aO'=l,  (aO''=l,  and  since 
(^^)^=  1,  (^^)-^=  1.     Therefore, 

io^y(.P'y  or  (a^l^'^y  or  (a^ySO"=l. 
Since  (a^P'y=l,  a^p^  is  a  root  of  jr"— 1=0.  Therefore, 
any  term  given  by  the  product  (1)  is  a  root  of  ^"—1=0. 
The  product  (1)  gives  7t  roots  of  jt''— -1  =  0,  and  hence 
(1)  gives  all  the  roots  of  ;i:"— -1=0  unless  two  roots  given 
by  (1)  are  equal.  Suppose  that  two  roots  given  by  (1)  are 
equal,  if  possible.     For  example,  suppose  . 

a^^^=a«^^ 

then  a^-^=p'-'. 

The  expression  on  the  left  side  of  this  equation  is  a  root 
of  ^—1=0,  and  the  expression  on  the  right  side  is  a  root 
of  ;t:'— -1=0.  But  by  Art.  790  these  expressions  cannot 
be  equal.  Therefore,  (1)  gives  all  the  roots  of  jj;"— 1  =  0. 
In  a  similar  way  we  show  that  if  n  is  the  product  of 
/>^r^^  prime  numbers,  r,  s,  and  /,  that  the  roots  of  ;r"— 1=0 
can  be  found  from  the  roots  ;v''— 1=0,  x'—l=0,  and 
:xf—l=^0,  and  so  on. 


SPECIAL  EQUATIONS.  6 1 3 

792.  In  general,  by  a    Primitive  nth  Root  of  +1 

is  meant  any  root  of  the  equation  ^i:''— 1=0  that  is  not  also 
a  root  of  a  binomial  equation  of  lower  degree.  Thus,  of  the 
roots  of  ;i;6-l=0,  ~|+i-i/:i3,  -i-^i/ITs,  and  +1, 
are  also  roots  of  the  equation  x^  — 1=0,  while  + 1  and  ~  1 
are  also  roots  of  the  equation  .r^  — 1  =  0.  This  leaves 
-|-+^l/— 3  and  ^—^V —S  as  primitive  roots  of  :r^  — 1=0. 
Likewise  we  speak  of  the  primitive  roots  of  x"-{  1  =  0. 

Plainly,  all  the  roots  of  ;t:"— 1=0  except  1  are  primitive 
if  71  is  a  prime  number.  For,  n  being  a  prime  number,  it 
is  prime  to  all  lesser  numbers,  and  therefore,  by  Art.  790, 
;*:"— 1  =  0  cannot  have  a  root  except  1  which  is  also  a 
root  of  a  binomial  equation  of  lower  degree. 

Whatever  value  n  may  have,  it  is  plain  that  the  second 
of  the  72th  roots  of  -f  1,  if  arranged  as  in  Art.  785,  is 
always  a  primitive  root  of  ;t'" — 1=0. 

793.  A  general  solution  of  the  binomial  equation  is 
given  by  Art.  784.  But  since  the  equations  ;i;"+l=0, 
and  :t:"— 1  =  0  are  reciprocal ^(^dXxoxvs,  they  maybe  solved 
algebraically  if  not  of  too  high  degree.  \in  is  a  large  num- 
ber, but  composed  of  several  factors  prime  to  each  other, 
then  the  binomial  equation  may  often  be  solved  by  re- 
ducing its  solution  to  the  solution  of  several  equations  of 
lower  degree,  as  indicated  in  the  last  article. 

Below  are  given  a  few  examples  of  the  solution  of 
binomial  equations: 

(1).    Solve  A'3-- 1=0. 

Removing  the  factor  x  —  \  from  this  reciprocal  equation,  we  have 

Solving  this  quadratic,  we  get 

Thus  x  =  l,-|  +  ^\/-3  and —J-jV^. 

Notice  that  if  we  call 

-^i^rW~^^-^*  then  --^-jv'^rrwz  ^nd  l=a)». 


6l4  UNIVERSITY    ALGEBRA. 

(2).     Solve  .r6 -1=0. 

We  may  reduce  this  to  the  standard  forjji  of  a  reciprocal  equation 
by  dividing  by  x"^ —  \,  and  then  solve;  or,  we  may  proceed  thus: 
Write  out  all  the  terms  of  (a  +  a^)(^  +  /5'^  +  ^^),  in  which  a  is  a 
primitive  root  of  jc^  — 1=0  and  p  is  a  primitive  imaginary  root  of 
x^  — 1=0.     This  gives  these  six  roots  of  Jir6_l— 0  : 

i-4x/:=B.  i+ix/^  -i+i^z:^.  -i-iv/33,  _i+i. 

(3).     Solve  ^9-1=0. 
Write  the  equation 

(;.')' -1=0, 
in  which  the  values  of  x^    are  w,  0)3  and  ^^ .     Whence,   the  roots  of 
.^^  —  1=0  are  the  roots  of  the  three  equations 

;tr3_o)=i0,      ;>:3-<o2rzO,      .;»:  3— 0)3^=0. 
Dividing  :x;^-l  by  x^ —^^  or  x^ —  \,  we  get 
j,:6_|_^3_j_x=0. 
This  equation  contains  all  the  roots  of  the  equations  ji;3_a)— 0  and 
;^3_o)2z=0  ;  that  is,  x^■\■x^^-\—^  contains  all  the  primitive  roots  of 
;»:9-l  =  0. 

EXAMPI^KS. 

Solve  the  following  equations : 

1.  ;r8~l=0.  4.  ;^;9  +  l  =  0. 

2.  x^-{-l=^,  5.  x-^^-l^^, 

3.  ;i:io__i=:0.  6.  ;r6  +  l=0. 

7.  Reduce  the  solution  of  ^j;*^— 1=0  to  the  solution 
of  a  cubic  equation. 

8.  If  <o,  0)2,  0)3  are  the  cube  roots  of  +1,  show  that 

o)-f-w^  +  <«>3  =  0. 

9.  Show  that  the  sum  of  all  the  n  ;zth  roots  of  + 1 
equals  zero. 

10.  Show  that  a^+b^-\-c^-'Zahc 

=  (a  +  ^-f  r)  («  +  o)<^4- («^^^)  (^  +  («^^  ^+ <*>^). 

11.  Resolve  x^ ■\- x"^ ■\- x'^- -\- x -^\  into  real  factors  of 
the  second  degree. 


SPECIAL  EQUATIONS.  615 

CUBIC  KQUATIONS. 

794.  We  have  seen  that  any  equation  can  be  trans- 
formed into  another  in  which  next  to  the  highest  power 
is  wanting;  so  that  any  cubic  equation  can  be  transformed 
into  another  in  which  there  is  no  term  of  the  second 
degree.  We  will  therefore  confine  our  discussion  of  cubic 
equations  to  the  form  x^-\-ax-\-b=0.  The  solution  given 
below  is  usually  called  Cardan's  Solution. 

795.  Algebraic  Solution  of  the  Cubic. 

In  the  equation 

x^  +  ax+b=0,  (1) 

let  x—y+2,  then 

K^y+zY'Vaiiy-\-2)  +  b=0. 
Hence,  y^  +  z^  +  {^yz+a)(iy+z)-\-b=0.  (2) 

The  only  restriction  placed  upon  the  two  unknown 
numbers  y  and  z  is  that  their  sum  equals  x,  i,  e. ,  a  root 
of  the  given  equation  (1),  and  as  two  restrictions  not 
inconsistent  with  each  other  can  be  placed  upon  two  un- 
known numbers,  we  will  place  upon  z  the  further  restric- 
tion that  Syz  +  a=0. 

Introducing  this  restriction  into  equation  (2),  we  obtain 
^3+^3  +  ^=^=0.  (3) 

But  from  Syz+a=0  it  follows  that 

a 

Substituting  in  (3),  we  have 

or  _y6  +  5^3_g=0.  (4) 

Whence,  since  (4)  is  a  quadratic  in  terms  of  y'^, 


=  -^-7"  =  — 2=FY 


2T__ 
b 


a" 


and  ^'  =  -^-7"  =  — o=FA/4-  +  2f 


6l6  UNIVERSITY    ALGEBRA. 

In  these  values  of  j/^  and  z^  we  must  take  the  upper 
signs  or  else  the  lower  signs  in  both  equations.  Select- 
ing the  upper  signs  in  both  equations,  we  have 

(      b   ^      ld\  a^  W     /      b         lb''   ,  a^  W 

796.  If  we  could  give  to  each  of  the  cube  roots  occur- 
ring in  the  value  of  x  any  one  of  the  three  values  of  which 
it  is  capable,  then  there  would  be  in  all  nine  values  of  x, 
whereas  there  should  be  only  three  values  of  x  since  an 
equation  of  the  third  degree  has  only  three  roots.  Evi- 
dently then,  these  different  cube  roots  must  be  combined 
in  some  way  so  as  to  give    only  three  instead  of  nine 

values.     From  the  equation  oyz-\-a=0,  we  get  y2:=  —  ^• 

Hence,  the  cube  roots  above  must  be  so  selected  that  their 

product  equals  —-x-    If  now  we  represent  one  of  the  cube 

roots  of  (  — ^-f-y/  T  +  9^  )  ^y  ^^^  ^^^  other  cube  roots  will 

be  mo)  and  ?n(t)'^  by  Art.  788,  and  if  we  represent  one  of 

b         Ib'^     a^ 
the  cube  roots  of  ■~o"~a/t'^27  ^^  ^^  ^^^  other  cube 

roots  will  be  thh  and  ;zo>2,  and  if  nt  and  n  be  so  selected 

that  mn=  —  77*  then  the  cube  roots  in  the  value  of  x  must 
o 

go  together  in  pairs,  thus : 

m  goes  with  n, 
moi  goes  with  nm'^, 
niiii^  goes  with  ;z(o, 
because  in  each  of  these  cases  the  product  is  the  same  as 
the  product  mn. 
We  may  then  write 

x=^ni-\-n, 
or  X'=m(ii-\-n(ji'^  ^ 

or  .r=Wa)2-f;^w. 


SPECIAL  EQUATIONS.  617 

797.  Let  us  now  examine  a  little  closer  into  the 
character  of  the  roots  of  a  cubic  equation.     The  values  of 

y^  and  ^^  given  in  Art.  795  are  both  real  i^  -j-+^  is  0 

or  any  positive  number,  and  imaginary  if  ^  +07  is  neg- 
ative. 

First.  Suppose -|- +07  is  positive.  Then  y^  and  2^ 
are  both  real  but  jk  and  2  may  be  either  real  or  imaginary. 
From  the  equation  y2=^  —-^  it  follows  that  these  quantities 

o 

y  and  2  are  either  both  real  or  both  imaginary.  Let  771 
and  71  denote  the  real  cube  roots  ofy^  and  2^  respectively, 
then  the  three  roots  of  the  given  equation  are 

m-}-7i, 
m(t)-\-7i(o'^y 
moi'^  -{-7i(D^ 

but  since                   a>=— i+i]/— 3, 
and                           co2  =  -^-^t/-3, 
therefore  the  three  roots  of  the  given  equation  are 
m-i-7i,  

—  i(77t  +  7l')-\-^(77Z-'7t')V  —  S 

—  ^lm  +  7l)  — 1(771  — 7l)l/  —  0, 

b"-     a^ 
Therefore,  when  ^  +9^  is  positive,  one  of  the  roots  of 

the  given  cubic  is  real  and  the  other  two  are  imaginary. 

Second.     Suppose  that  ^  +  ^^=0.  Inthiscase  theform 

of  the  roots  of  the  given  cubic  may  be  deduced  from  the 
preceding  case ;  for  now  ;?2=/z,  each  being  equal  to  the 

real  cube  root  of  -^«  The  three  roots  of  the  given  cubic  in 

this  case  are    therefore,  2/;^,   — 771,   —771.     Hence,  when 

^2     a^ 

-J +27  =  0  all  three  roots  of  the  given  cubic  are  real,  but 

two  of  them  are  equal  to  each  other. 


6l8  UNIVERSITY   ALGEBRA. 

Third.  Suppose  -j-  +  ^^  is  negative.  In  this  case,  be- 
cause y^  and  z^  are  imaginary,  y  and  z  must  also  be 
imaginary,  and  since  y^  and  z^  differ  only  in  the  sign  of 
the  imaginary  part  one  of  the  three  possible  values  of  y 
is  the  conjugate  of  one  of  the  three  possible  values  ot  z, 
Let  the  three  possible  values  of  y  be  represented  by 

r-\-is, 

then  the  three  possible  values  of  z  are 

r^is, 
(r — is)oi, 
(r — is)oi'^ . 
The  values  of  y  and  z  must  be  selected  in  such  a  way 

that  their  product  equals  —  k-'  which  is  real.     Hence,  if 

r+is  be  selected  as  the  value  of  j/,  then  r—is  must  be 
taken  as  the  value  of  z.  Since  this  is  the  only  one  of  the 
three  possible  values  of  z,  which,  multiplied  by  the 
selected  value  of  y,  makes  the  product  real.  Hence,  as 
in  the  preceding  article,  the  three  roots  of  the  given  cubic 
are 

r-\-is+r — is  or  2r,  _ 

(r-\-is)(o-\-(r—is)o>'^  or  — r— ^F  3. 
(r+is)(ii'^  +  (r — is^di  or  —r-{-sv^. 

Thus  we  see  that  when  ^^  +  07  is  negative  the  three  roots 

of  the  given  cubic  are  real  and  unequal. 

798.     We  have  found  that  when  -7  +  07    is    positive, 

the  equation  x^  +ax  +  b=^0  has  one  real  and  two  imagin- 
ary roots.  In  this  case  the  formula  enables  us  to  find  the 
real  root,  after  which  the  equation  may  be  reduced  to  one 

of  the  second  degree  which  may  readily  be  solved. 

^2      ^3 

We  have  also  found  that  when  -r  +  :7-;  is  negative,  the 

4      27 


SPECIAL  EQUATIONS.  619 

equation  x^-\-ax-\-d=0  has  three  real  roots,  two  of  which 
are  equal  to  each  other.  In  this  case  the  formula  enables 
us  to  find  all  three  roots. 

We  have  also  found  that  when  ^  +  ^—  is  negative,  the 

equation  x^-{-ax-i-d=0  has  its  three  roots  real  and  un- 
equal. In  this  case  the  expressions  above  given  for  y^ 
and  z^  are  each  imaginary,  and  although  we  know  that 
these  expressions  have  cube  roots,  there  is  no  arithmet- 
ical method  of  finding  them  and  no  algebraic  method  of 
finding  them  exactly.  In  this  case,  therefore,  although 
the  roots  are  real  and  distinct,  they  are  presented  to  us  in 
a  form  which  is  very  inconvenient  for  use.  For  this 
reason  this  case  is  often  referred  to  as  the ' '  irreducible  case, ' ' 

799.  We  have  stated  that  there  is  no  algebraic  method 
of  finding  exactly  the  cube  root  of  an  expression  of  the 

form  p+iq.     But  in  such  an  expression  as  (,p-{-iq)^,  if 

q<.p  w^e  ma}^  expand  by  the  binomial  theorem  into  a 

converging   series   arranged  according  to  the  ascending 

i_ 
powers  of  iq  and  find  an  approximate  value  of  {p-\-iq^^ 

in  the  form  P-\-  iQ,  where  P  is  the  sum  of  the  real  terms 

of  the  expansion,  and  iQ  is  the  sum  of  the  imaginary 

terms  of  the  expansion. 

If,  however,  ^>/>,  then,  since 

p^-iq-=i{q—ip), 
therefore  {p-\-iq)'^—i'^{(j—ipy^, 

and  since         i^-= — i  .'.   (—i)^  =  i.'.   —z=t^ 
therefore  (p+iq)'^=—iCq—ip)'^. 

Now,  —i(q—ip')'^  can  be  developed  by  the  binomial 
theorem  into  a  converging  series  arranged  according  to 
ascending  powers  of  ip,  and  from  this  development  an 
approximate  value  of  —iiq—ip)^  can  be  found  as  in  the 
preceding  case. 


620  UNIVERSITY   ALGEBRA. 

KXAMPI^KS. 

1.  Given 

(2  +  111/^)^=2  +  1/^ and  (2-111/^)  3  =  2-1/^ 
solve  the  equation  ;r^  — 15jt:— 4=0. 

2.  Given 

(10+ V'108) "^=1  +  1/3  and  (l0-l/l08)^=l-l/3 
solve  the  equation  jr-^  +  6x— 20=0. 

3.  Solve  the  equation  x^  —  24iX+72  =  0, 

4.  Show  that  if  a  is  any  positive  number,  the  equa- 
tion x^-i-ax-i-d=0  has  two  imaginary  roots. 

21/ "3 

5.  Show  that  the  equation  x^ — c'^x-\ ^^=0  has 

two  roots  which  are  equal  to  each  other  whatever  be  the 
value  of  c. 

6.  Show  that  the  equation  .r^  — 12;r+3=0  has  its 
three  roots,  real  and  unequal,  when  d  is  numerically  less 
than  16,  and  has  two  imaginary  roots  when  d  is  numeri- 
cally greater  than  16. 

7.  Show  that  the  equation  x^+ax+d=0  has  two  im- 
aginary roots  when  a  is  any  positive  number  whatever. 

BIQUADRATIC  EQUATIONS. 

800.  Since  an  equation  can  always  be  deprived  of  the 
term  of  next  to  the  highest  degree,  we  confine  our  dis- 
cussion of  biquadratic  equations  or  equations  of  the  fourth 
degree  to  the  form 

x^-\-ax^-  +  dx+c=0. 

The  solution  which  follows  is  called  Descartes'  solution. 

Suppose  x'^+ax^-j-dx+c-  (x'^  -j-Ax-{-B')(ix^-  +  Cx  +  B) 
where  A,  B,  C,  and  D  are  undetermined  coefficients. 


SPECIAL  EQUATIONS.  62  I 

Then  x"^ +ax'^-\-dx-\-c 

=x^-i-(A  +  C)x^  +  CB+AC+D')x^  +  (iAD+BC)x+BD. 
Equating  coefficients : 

A-hC=0        whence  A^-C 

B+AC+B=a  ''         B+D-A^  =  a 

An+BC=d  ''  A{^D-B)  =  b 

BD=c  *'  BD=c 

Therefore 

AD-\-AB=A^  +  aA]  AD-AB^b;  A'^BD=cA'^. 
Finding  AB  and  AD  from  the  first  two  equations  and 
substituting  in  the  third, 

{A^-\-aA-^)(iA^-\-aA-b)  =  4.cA'^. 
Multiplying  out,  recognizing  a  product  of  sum  and  dif- 
ference, 

A^  +2aA^  -\-(a''  -ic)A'^—b''=0, 

which  is  a  cubic  in  terms  of  y^^.  Therefore  A  can  be 
found  and  then  B,  C,  and  D  from  the  equations  above. 
Finally,  four  values  of  x  can  be  found  by  solving  the  two 

quadratics. 

x'^+Ax-^B=0  and  x'^  +  Cx+D=0. 

801.  We  notice  that  the  equation  for  finding  A  is  of 
the  sixth  degree,  but  that  this  is  what  we  ought  to  ex- 
pect may  be  readily  shown.  If  we  represent  the  roots  of 
the  given  equation  by  r^,  r^,  ^3,  ^4,  we  know  that 

x^-\-ax'^-\-bx-]-c=(^x—r^'){x—ro)(x—'r^^{x—r^)  (1) 
also    x'^^-ax'^-{-bx^-c=(^x''--\-Ax+BXx'^^Cx+D)     (2) 

From  (1)  and  (2)  it  is  evident  that  x'^+Ax+B  is  the 
product  of  two  of  the  linear  factors  of  the  second  membe^ 
of  (1),  and  as  a7ty  of  these  four  factors  may  be  taken 
together,  and  as  two  things  can  be  selected  from  four 
things  in  six  ways,  therefore  x'^+Ax+B  may  stand  for 
any  one  of  six  quadratic  expressions.  But  from  the 
solution  given  above  it  is  evident  that  when  A  is  fixed 
B  is  also  fixed,  and  hence  when  A  is  fixed  the  whole 


622  '  UNIVERSITY   ALGEBRA. 

expression  x'^+Ax+B  is  fixed.  Now,  if  x'^  -]-Ax-\-B 
may  stand  for  any  one  of  six  quadratic  expressions,  and 
if  this  expression  is  fixed  when  A  is  fixed,  A  ought  to 
have  six  values;  /.  e.,  A  ought  to  be  found  from  an 
equation  of  the  sixth  degree. 

Moreover,  it  is  readily  seen  from  the  solution  given 
above,  by  computing  the  values  of  ^,  C,  and  D  in  terms 
of -^,  that  changing  the  sign  of  A  merely  interchanges 
the  tw^o  quadratic  factors  in  the  second  member  of  (2); 
so  that  when  one  quadratic  factor  corresponding  to  a 
certain  value  of  A  is  found,  another  factor  may  be  found 
by  simply  changing  the  sign  of  A.  Hence,  of  the  six 
values  of  A,  three  are  simply  the  negatives  of  the  other 
three.  Thus  we  see  why  the  equation  from  which  A  is 
found  is  a  aidic  in  terms  of  A"^.  This  cubic  is  called  the 
Auxiliary  Cubic. 

802.  Let  us  now  notice  the  relation  between  the  roots 
of  the  given  biquadratic  and  those  of  the  auxiliary  cubic. 

If  the  roots  of  the  biquadratic  are  all  real,  then,  since 
A  is  the  sum  of  fwo  of  these  roots  with  their  signs  changed, 
A  is  real;  therefore,  A'^  is  real;  therefore,  the  roots  of 
the  auxiliary  cubic  are  all  real. 

Again,  if  the  roots  of  the  given  biquadratic  are  all  im- 
aginary, since  their  sum  equals  zero  they  must  have  the 
forms  a+?yS,  a— /y8,  — a+zy,  — a— /y,  and  as  A  is  the  sum 
of  hi^o  of  these  roots  with  their  signs  changed,  therefore 
the  only  possible  values  of  ^  are  db2a,  db/(/3+y),  =h2*(/5— y), 
therefore  the  only  values  of  A  ^  are 

4a2,  -(^+y)^  -(/?-y)^ 
which  are  all  real. 

Thus  we  see  that  when  the  roots  of  the  given  biquadratic 
are  all  real  or  all  iviaginajy  the  auxiliary  cubic  falls  ujider 
the  irreducible  case  unless  this  cubic  has  equal  roots. 


SPECIAL  EQUATIONS.  623 

If  the  given  biquadratic  has  two  real  and  two  imaginary 
roots,  then,  since  the  sum  of  the  roots  equals  zero,  the 
roots  must  have  the  forms 

a  +  //3,      a—//?,       —a  +  y,       — a— y ; 

therefore,  the  only  possible  values  of  ^  are  d=2a,  d=(y+2^), 
db(y— //?);  therefore,  the  only  possible  values  of  A"^  are 

4a2,  (a^—l3'^)  +  2z/3y,  a'^  —  p'- —2iPy ; 
and  if  y^O  two  of  these  values  are  imaginary  and  one 
real,  but  if  y=0  all  three  values  of  ^^  are  real  but  two  of 
them  are  equal  to  each  other.  But  when  a  cubic  equation 
has  three  real  roots,  two  of  which  are  equal  to  each  other, 
or  one  real  and  two  imaginary  roots,  the  equation  does 
not  fall  under  the  irreducible  case.  Therefore,  whe7i  the 
given  biquadratic  has  two  real  and  two  imaginary  roots^ 
the  auxiliary  cubic  does  not  fall  under  the  irreducible  case. 

803.  The  roots  of  the  biquadratic  may  be  expressed 
in  terms  of  the  roots  of  the  auxiliary  cubic. 

Let  a^,  /32^  yi  represent  the  three  roots  of  the  auxiliary 
cubic,  then  we  have 

a2/?2^'^  =  ^^  (1) 

a24-;S2-fy2  =  -.2^.  (2) 

Now,     x''-\-Ax^B^x''-^Ax-^\[A''-^a-^' 

Substituting  herein  the  values  of  a  and  b  found  from 
(1)  and  (2)  and  taking  a  for  the  value  of  y^,  we  obtain 
x''  +Ax-\-B=x''  +a;ir+i[a2  -^(a^  +^2  _^y  2)_^^] 

=  ;^2_|.^^^^(-^2_^2_^2_2/3y) 

Placing  this  last  expression  equal  to  zero  and  solving, 
we  get 

^=-i(a+iS+y)  orK~«+^+r). 
Treating  the  vSecond  quadratic  factor  x'^-]-Cx+D   in   a 
similar  manner,  we  obtain 

^=-K^-/5+7)  or  x(a+^^y). 


624  UNIVERSITY    ALGEBRA. 

Hence,  the  four  roots  of  the  given  biquadratic  equation 
are 

—K^+P+y),  K-^+^+y),  K^-P+y).  K^+P-y)- 

EXAMPLES. 

1.  Solve  the  equation  x"^ — 5;r"  +  10;i;— 6=0. 

2.  Solve  the  equation  x^—x'^-{-2x+2, 

3.  Solve  the  equation  jr^  — 9:^2-— 4%'4- 12=0. 

4.  Solve  the  equation  x'^—6x'^+Sx—S=0. 

5.  Solve  the  equation  x"^  —  4x^  -\-16x—16=0, 

6.  From  the  relation  between  the  roots  of  a  biquad- 
ratic equation  and  its  auxiliary  cubic  equation,  express 
the  roots  of  the  cubic  in  terms  of  those  of  the  biquadratic. 

7.  If  a  biquadratic  equation  has  one  root  equal  to  zero 
and  has  no  term  in  ^r^,  show  that  the  roots  of  the  auxil- 
iary cubic  are  the  squares  of  the  three  remaining  roots  of 
the  biquadratic. 

8.  If  a  biquadratic  equation  has  three  equal  roots  and 
has  no  term  in  x^,  show  that  the  auxiliary  cubic  has  two 
equal  roots. 

9.  If  a  biquadratic  equation  has  three  equal  roots  and 
has  no  term  in  x^,  show  that  the  auxiliary  cubic  has 
three  equal  roots  and  hence  is  a  perfect  cube. 

10.  If  a  biquadratic  equation  has  three  of  its  roots 
proportional  to  the  numbers  1?  2,  3  and  has  no  term  in 
x'^^y  show  that  the  roots  of  the  auxiliary  cubic  are  propor- 
tional to  the  numbers  9,  16,  25. 


SPECIAL  EQUATIONS.  625 

Historical  Note.  — Of  the  few  discoveries  in  pure  mathematics 
by  the  Arabs,  the  most  creditable  is  their  geometric  solution  of  cubic 
equations.  At  the  beginning  of  the  eleventh  century,  Abul  Gud 
made  much  progress  in  the  mastery  of  this  problem.  Soon  after,  it 
was  brought  to  a  more  complete  solution  by  Omar  al  Hayyami.  The 
roots  were  determined  by  the  intersection  of  conic  sections.  Only 
positive  roots  of  the  equations  were  recognized  in  these  constructions. 

The  algebraic  solution  of  cubics  was  discovered  in  Italy  in  the  six- 
teenth century.  The  first  to  attack  the  problem  was  Scipio  Ferro, 
but  his  mode  of  solution  was  never  made  public.  It  was  the  practice 
in  those  days  to  keep  discoveries  secret,  to  secure  by  that  means  an 
advantage  over  rivals  by  proposing  problems  beyond  their  reach. 
The  first  solution  of  cubics  handed  down  to  us  is  that  of  Tartaglia. 
(died,  1557).  In  a  contest  with  a  pupil  of  Ferro,  he  beat  him  by 
solving  thirty  cubic  equations  in  two  hours,  while  Ferro' s  pupil  could 
not  solve  any  of  those  proposed  by  him.  Tartaglia' s  victory  became 
known  to  Hieronimo  Cardano  (1501-1576)  of  Milan,  who,  after  giving 
the  most  sacred  promises  of  secrecy,  succeeded  in  obtaining  from 
Tartaglia  a  knowledge  of  his  rules.  But  Cardan  broke  his  solemn 
pledge  by  inserting  the  much  sought  for  rules  in  his  A7's  Magna. 
Tartaglia  had  intended  himself  to  publish  a  work  of  which  his  solu- 
tion of  cubics  should  be  the  crown.  Cardan's  treachery  made  Tar- 
taglia desperate.  A  long  and  acrimonious  controversy  between  the 
two  parties  followed,  in  which  Tartaglia  demonstrated  his  superiority 
as  a  mathematician.  Modern  writers  have  done  Tartaglia  great 
injustice  by  attributing  the  solution  of  cubics  to  Cardan. 

To  one  of  Cardan's  pupils,  Ferrari,  belongs  the  honor  of  the  general 
solution  of  equations  of  the  fourth  degree.  The  solution  was  published 
by  Bombelli,  in  his  treatise  on  algebra,  in  1579.  A  solution  known  as 
Simpson's,  which  is  not  essentially  different  from  Ferrari's,  was  pub- 
lished about  1740.  In  1637,  Descartes'  work  appeared.  Besides  his 
solution  of  the  biquadratic  equation  it  contained  many  important  ad- 
ditions to  algebra,  especially  the  recognition  of  negative  and  imaginary 
roots  of  equations  and  the  "  Rule  of  Signs"  which  is  given  in  the  next 
chapter. 

Descartes  attempted  to  obtain  a  general  algebraic  solution  of  equa- 
tions but,  of  course,  failed.  Euler  attempted  the  same  problem  with 
the  same  result,  and  throughout  the  eighteenth  century  many  math- 
ematicians busied  themselves  with  this  problem,  but  all  attempts 
failed  for  equations  above  the  fourth  degree. 

On  account  of  the  failure  to  solve  general  equations  beyond  the 
fourth  degree,  it  was  natural  that  mathematicians  should  query 
whether  such  solutions  were  possible.  Demonstrations  have  been 
given  by  Abel  and  Wantzelof  the  impossibility  of  ^oWm^  algebraically 
a  general  equation  above  the  fourth  degree.  A  transce^tdental  solution 
of  an  equation  of  th.3  fifth  degree  has  been  given  by  Hermite  in  a  form, 
involving  elliptic  integrals. 

40  -  U.  A. 


CHAPTER  XXXIV. 


SKPARATION   OF   ROOTS. 


804.  As  we  have  already  seen,  equations  of  the  first, 
second,  third  and  fourth  degrees  can  be  solved  algebraically. 
By  this  we  mean  that  when  an  equation  has  literal  coeffi- 
cients and  is  perfectly  general  in  form,  there  is  a  straight- 
forward process  of  solution  which  will  enable  us  to  find 
the  roots,  and  that  the  roots  will  be  expressed  in  terms  of 
the  coefficients  in  such  a  manner  as  to  involve  only  the 
ordinary  operations  in  algebra,  viz :  addition,  subtraction, 
multiplication,  division,  and  involution  and  evolution  to 
commensurable  powers  and  roots. 

Abel,  a  Norwegian  mathematician,  has  proved  that  a 
general  equation  above  the  fourth  degree  cannot  be  solved 
algebraically;  that  is,  the  roots  cannot  be  expressed  by 
means  of  the  ordinary  symbols  of  operation  alone,  but 
the  function  of  the  coefficients,  which  expresses  the  roots, 
is  not  within  the  range  of  algebraic  analysis. 

Although  we  cannot  solve,  algebraically,  a  general 
literal  equation  above  the  fourth  degree,  still  methods 
are  known  which-enable  us  to  find  the  roots  of  nuuiericat 
equations  above  the  fourth  degree,  and  these  methods  it 
will  be  the  object  of  this  and  the  following  chapter  to 
explain. 

805.  The  solution  of  numerical  equations  embraces 
two  distinct  problems  ;  first,  the  separation  of  roots,  and 
second,  the  numerical  calculation  of  the  roots.  The  first 
of  these  two  problems  is  the  only  one  we  shall  consider 
in  this  chapter. 


SEPARATION   OF    ROOTS.  627 

8C6.  Any  of  tlie  roots  of  an  equation  are  said  to  be 
separated  from  the  other  roots  when  two  numbers  are 
found  between  which  these  roots  and  no  others  lie.  Our 
attention  at  first  will  be  confined  to  real  roots,  and  of 
these  we  shall  first  consider  the  separation  of  the  positive 
from  the  negative  roots. 

807.  A  Superior  Limit  of  the  positive  roots  of  an 
equation  is  any  positive  number  greater  than  the  greatest 
of  the  positive  roots. 

An  Inferior  Limit  of  the  positive  roots  of  an  equation 
is  any  positive  number  less  than  the  least  of  the  positive 
roots. 

A  sitperiof  limit  of  the  negative  roots  of  an  equation  is 
any  negative  number  numerically  greater  than  the  numeri- 
cally greatest  of  the  negative  roots. 

An  inferior  limit  of  the  negative  roots  of  an  equation  is 
any  negative  number  numerically  less  than  the  numeri- 
cally least  of  the  negative  roots. 

808.  If  tivo  numbers  be  substituted  in  turri  for  x  in  f{pc) , 
giving  results  with  opposite  signSy  there  is  an  odd  7iumber 
of  real  roots  of  fix)  between  the  numbers  substituted. 

Suppose  a</?  and  suppose  f{x)  is  positive  when  x=(x 
and  negative  when  x—^,  then,  as  x  changes  continuously 
from  a  to  /?,  fix)  changes  from  positive  to  negative ;  and 
as/(;^;)  is  continuous  (Art.  748)  it  must  pass  through  zero 
when  it  changes  sign;  that  is,' /(;t:)  must  become  zero 
for  one  or  more  values  of  x  between  a  and  ^;  that  is,  fix) 
has  one  or  more  real  roots  between  a  and  ^, 

The  same  argument  evidently  applies  \i  fix)  is  nega- 
tive when  x=a  and  positive  when  x=p. 

We  thus  see  that  /(:*:)  has  one  or  more  real  roots  between 
a  and  /3,  but  it  is  easily  seen  that  the  number  of  roots  must 
be  odd;    for,  every  time  the  function  changes  sign,  no 


628  UNIVERSITY    ALGEBRA. 

matter  whether  the  change  is  from  positive  to  negative  or 
from  negative  to  positive,  there  must  be  a  real  root  of/(jt:); 
but  if,  as  X  changes  from  a  to  ^,  the  function  begins  with 
being  positive  and  ends  with  being  negative,  it  must  have 
changed  sign  some  odd  number  of  times.  The  same  is 
true  if  the  function  begins  with  being  negative  and  ends 
with  being  positive,  and  hence  there  must  be  an  odd  num- 
ber of  real  roots  between  a  and  /3. 

809.  It  is  now  easily  seen  that  a  superior  limit  of  the 
positive  roots  of  the  equation  f {%)={)  is  a  number  which 
gives  to  f{pc)  a  sign  which  cannot  be  changed  by  increas- 
ing X  beyond  this  superior  limit.  For,  if  A  represents  a 
superior  limit  of  the  positive  roots  of  /(^)  =  0,  and  if  we 
suppose  /(A)  is  positive,  then  a  being  any  number  greater 
than  X  if  it  were  possible  for  /(X)  to  be  negative,  there 
would  be  a  real  root  of /(;r)=0  between  A  and  a  and  \ 
would  not  be  a  superior  limit  of  the  positive  roots. 
Therefor^,  f{x)  cannot  be  negative.  Therefore,  f{pc) 
cannot  change  sign  by  increasing  x  from  the  value  A. 
Also,  if  A  represents  an  inferior  limit  of  the  positive  roots 
of /(^)=0,  and  if  a  be  any  positive  number  less  than  A, 
it  is  evident  that/(^)  and /(a)  must  have  the  same  signs. 

A  similar  remark  applies  to  both  inferior  and  superior 
limits  of  the  negative  roots.  Conversely,  if  f{pc)  always 
preserves  the  same  sign  for  any  value  of  x  equal  to  or 
greater  than  x^  then  evidently  x\s  2i  superior  limit  of  the 
positive  roots  of/(:t:)=0;  and  if  y*(^)  always  preserves 
the  same  sign  for  any  value  of  x  equal  to  or  less  than  x^ 
then  ;fis  an  inferiorXxxmX,  of  the  positive  roots  of /(;t:)=0. 

Upon  the  principles  just  given  we  will  found  the  dis- 
cussion of  the  limits  of  the  roots  of  equations.  We  will 
confine  our  attention  at  first  to  the  superior  limits  of  the 
positive  roots. 


SEPARATION   OF   ROOTS.  629 

810.     /^^  the  equation 

^"'+/>i^"-'+/'2-^"~'+  •  •  •  +A=0, 
if — N represent  the  7iumerically  greatest  negative  coefficient^ 
N-\- 1  is  a  superior  limit  of  the  positive  roots. 

lyet  the  first  member  of  the  given  equation  be  repre- 
sented by  f{x').  Then,  evidently,  any  positive  value  of 
X  which  makes 

x''—N{x''-'^-Vx''-'^^-  .  .  .  +1) 

positive  will  also  make  f(^x)  positive. 

But      ;r«-A^(;r'^-i+;t:«-2+  .  .  .  +l)=x''-A^ 


«_1 


x-\ 
-1        ...       ,.^"-1 


and  x''-Kf^^—^>x''-\-N- 


also 


x—\  x—1 

Therefore,  any  positive  value  of  x  which  makes 

positive  will  also  make  f(x')  positive. 

/         N  \ 
But  (;t;"— 1)(  1 —TT )  is  evidently  positive  if  x—l';>JV; 

that  is,  if  ;t:>iV^+l. 

Therefore,  f^x)  is  positive  for  any  value  of  x  which  is 
greater  than  N+ 1. 

Again,  if  x=N+l,  then 

X**  —  1 

x''--N^^—i=l. 
x—1 

Hence,  x*'—N(ix*'~'^+x''-''^-{'  •  •  •  +1)   is  positive,  and 

therefore /(;»;)  is  also  positive  when  x=N+l.  Therefore, 

when  X  is  equal  to  or  greater  than  A^+l,/(-^)  is  positive. 

Therefore,  N+1  is  a  superior  limit  of  the  positive  roots 

of/(;t)=0. 


630  UNIVERSITY    ALGEBRA. 

811.     In  the  equation 

if  pr  be  the  first  7iegative  coefficient^  ajid  if  the  nnmerically 
greatest  negative  coefficient  be  represe?ited  by  — N^  theJi 
1  +  V N  is  a  superior  li?nit  of  the  positive  roots. 

Let  us  represent  the  first  member  of  the  given  equation 
by  f{pc),  then  for  any  positive  value  of  x  each  term  before 
the  term  piX''~''  is  positive.  Hence,  for  any  positive 
value  of  X 

/(:r)>;^«+/^^^-''+A+i-^''~''"'  +  '  '  *  +A.  (1) 

.-.    /(;r)>:t:"-iV(;r«-''+;»;"-'-i+  .  .  .  +1).  (2) 

i.  e. ,  /(^)>^«~i\^^!^^.  (3) 

Hence,  for  any  value  of  x  greater  than  1, 

Therefore,  f{x)  will  be  positive  for  any  value  of  x  that 
satisfies  the  two  inequalities 

x>\  and;r"(^— 1)— iV;«^""'"^^>0. 
The  second  of  these  inequalities  may  be  written 

;i;''-i(:r-l)-A^>0, 
which  is  evidently  satisfied  when 

{x-\r-N>^, 
that  is,  when  x—V>1/N, 

that  is,  when  x>  1  +  V N. 

Therefore,  f{x)  is  positive  for  any  value  of  x  which  is 
greater  than 

_     \^Vn. 

Also  if  ;r=  1  + "!/ N  the  inequality 

x"-!  (:*:■- 1)~A^>0 
is  still  satisfied,  for,  x  being;  positive, 
;f''-i(^-lj>(^-iy, 


SEPARATION    OF    ROOTS.  63 1 

and  since  in  this  case  A^=(;tr— 1)'',  therefore 

Therefore,  fix)  is  positive  when  x^=^\-\-V N. 

Therefore,  any  value  of  x  equal  to  or  greater  than 
\-\-V N,  makes  /(;t)  positive,  and  hence  \-\-V N  is  a 
superior  limit  of  the  positive  roots  of/(;i:)=0. 

812.  If^  2;^  a  ratio7ial  integral  function  of  x^  s(^y  f(.^^y 
each  negative  coefficient  be  divided  by  the  sum  of  all  the  pos- 
itive coefficients  which  precede  it^  and  if  the  nuTnerically 
gjxatest  quotient  thus  formed  equals  —  iV,  then  N+\  is  a 
superior  limit  of  the  positive  roots  of  f{x)=^0. 

Suppose 
f(^x)  =  a^x''  +  a^x''-^  +  a^x"-^--a,^x''-'^+  •  .  .  +^„, 
where  a^,  a^,  a^,  «3,  etc.,  are  all  positive  numbers,  and 
a  negative  coefficient  whenever  it  occurs  is  indicated  by 
the  presence  of  a  minus  sign. 

x—1 

.'.     x''-l  =  (_x-l)(x'-'^  +X'-'' -{-x"-^  +  .  .  .  +1). 

x''=lx'-l)Cx''-^  ^x"--^  +x''-^  +  ■  .  .+1)  +  1. 
Now  let  us  use  this  equation  as  a  formula  by  which  to 
transform  each  term  of  f(x')  in  which  the  coefficient  is 
positive,  and  leave  the  terms  in  which  the  coefficients 
are  negative  without  change.     We  thus  obtain  an  ex- 
pression for  /(jir)  arranged  as  follows  : 
aQ(^x—l)x"~^+aQ(x^l)x"-''^  +  aQ(x—l)x''~^-j-  •  .  .  -j-a^ 
a^lx—l)x"-'^  +a^(x--l)x''-^  +  ■  .  .  +a^ 
a2(x—l)x''-^-i-  .  .  .+^2 
^a^x''"^ 

+  .  .  . 
In  this  arrangement  it  is  to  be  noticed  that  all  the  terms 
in  any  one  horizontal  line  come  from  a  single  term  of/(;i;), 
the  first  horizontal  line  coming  from  the  first  term  of/(jr), 


632  UNIVERSITY    ALGEBRA. 

the  second  horizontal  line  from  the  second  term  oif{x), 
etc.;  and  since  the  fourth  term  of /(:t:)  has  a  negative 
coefficient  and  therefore  is  left  unchanged,  the  fourth 
horizontal  line  in  this  arrangement  consists  of  but  a  single 
term,  and  the  same  is  evidently  true  of  any  other  hori- 
zontal line  which  comes  from  a  term  of /(:r)  in  which  the 
coefficient  is  negative.  Moreover,  a  little  reflection  upon 
the  way  in  which  the  above  expression  is  arranged  will 
make  it  clear  that  when  a  negative  coefficient  appears,  it 
must  occur  at  the  bottom  of  some  vertical  column  and  that 
only  one  negative  coefficient  can  occur  in  any  one  column. 
Now  let  us  consider  the  successive  vertical  columns  in 
the  above  arrangement.  When  no  negative  coefficient 
occurs  in  a  column,  the  value  of  that  column  is  evidently 
positive  when  x^\.  To  insure  a  positive  value  to  the 
third  column,  we  must  have 

(^o+^i+«2)(-^"— l)>^a) 


0+^1+^2 

a. 


i.  e.y  x>li--      . 

Similarly,  ii  f{x)  contains  a  term  —  ^;,:r''~'' this  term 
appears  in  the  above  arrangement  at  the  foot  of  the  rth 
vertical  column  and  the  terms  above  this  in  the  same 
column  contain  onl}^  the  positive  coefficients  of/(-r)  which 
precede  the  term  —arX"~*\  Therefore,  that  the  value  of 
the  ;'th  column  may  be  positive  we  must  have,  by  reason- 
ing as  in  the  case  of  the  third  column, 

<^0+^l+^2+<^4+  •    •   • 

where  the  denominator  is  the  sum  of  the  positive  coeffi- 
cients which  precede  the  term  —  a^""''. 


SEPARATION    OF    ROOTS.  633 

Now  there  may  be  several  terms  in  f{pc)  which  have 
negative  coefficients,  and  therefore  several  columns  in  the 
above  arrangement  in  each  of  which  a  negative  coefficient 
appears  at  the  foot  of  the  column,  and  we  now  see  how  x 
can  be  taken  in  such  a  way  as  to  render  the  value  of  any 
one  of  these  columns  positive.  Therefore,  if  x  be  taken 
greater  than  the  greatest  of  the  expressions  of  the  form 

1-1 ; -, ,  the  value  of  each  column  in  the  above 

^o+<^i+  •  •  • 
arrangement  is  positive,  and  hence  the  sum  of  all  these 

columns  or  f{^x)  is  positive.     If  x  be  taken  equal  to  the 

greatest  oi  Wi^  expressions  of  the  form  1-| — — ^- j 

the  value  of  one  of  the  columns  in  the  above  arrangement 
is  zero,  but  each  of  the  others  is  positive,  therefore  the  sum 
of  all  the  columns  or  fix)  is  positive.  Therefore,  f{pc) 
is  positive  for  any  value  of  x  equal  to  or  greater  than  the 

greatest  of  the  expressions  of  the  form  \-\ — — "— 

Therefore  the  greatest  of  the  exjjressions   of  the   form 

1-1 "L is  a  superior  limit  of  the  positive  roots 

a^-\-a^-\-  •  •  • 
of/(x)==0. 

813.  Newton's  Method.  If  a  number,  say  /;,  be 
chosen  so  that  f{x)  and  its  successive  derivatives  are  all 
positive  when  x=h,  then  h  is  a  superior  li?nit  of  the  positive 
roots  of  the  equation  f(x)=0. 

Let  f(x)  =  x''+p,x''-'+p^x''-'-+p^x''-^+-  .  .  +/,,and 
let  the  equation  f(x')=0  be  transformed  by  substituting 
jy-{-h  {or  X.     By  Art.  747  /(^)=0  becomes 

L_  L  L_ 

Now,  if  the  coefficients  of  the  various  powers  of  y  are 
positive,  it  is  evident  that  there  can  be  no  positive  value 
of  jK  that  will  satisfy  this  equation,  and  My  cannot  be 


634  UNIVERSITY   ALGEBRA. 

positive  X  cannot  be  greater  than  h,  therefore  ^  is  a 
superior  limit  of  the  positive  roots  of  the  equation 
/(^)=0. 

The  problem  now  is  to  find  the  number  h  which,  when 
put  for  X,  will  render  f{pc)  and  its  various  derivatives 
positive. 

The  method  of  finding  h  will  be  understood  from  a 
special  case.     Let  us  take  the  the  equation 

;t:5__5^.4_5^3_^25jt;2+4;r— 20=0. 
Here  f{^x)^x^-'hx^  —  hx'^-\-1hx'^-\-A:X—%) 

/'(x')  =  5x^—20x^-16x^-{-50x+4: 
/^\x)  =  20x'''-60x''-S0x+50 
/'^'(x)  =  mx''-120x 
/^X-^)  =  120;»;-120 
/\x)  =  120. 
•    f^^x)  is  positive  independent  of  the  value  of  x. 
f'^X^)  is  positive  when   x=2. 
f^'ipc)  is  negative  when  :r=2. 
f'ipc)  is  positive  when   .r=3. 
f"(x)  is  negative  when  ;r=3. 
f"{x)  is  positive  when   jr=4. 
f\x)  is  negative  when  x=4:. 
f'{x)  is  positive  when  ;r=5. 
fix)  is  zero  when  x=^h. 

fix)  is  positive  when  jt:=6. 
Now,  if  the  successive  derivatives  be  examined  when 
^=6,  it  will  be  found  that  they  are  all  positive;  therefore, 
6  is  a  superior  limit  of  the  positive  roots  of  the  equation 
/(^)=0. 

In  this  work  it  will  be  noticed  that  we  began  with  the 
last  function  containing  x)  viz.:  f^ix),  and  found  the 
smallest  positive  integral  value  of  x  which  makes  this 
function  positive,  then  substituting  this  value  in  the  next 
preceding  function,  viz.,  f"\x)  and  finding  in  that  case 


SEPARATION   OF   ROOTS.  635 

f"\x)  is  negative  we  increased  the  value  of  x  by  unity. 
If  the  value  of  x,  thus  increased,  had  rendered  f"\x) 
negative,  we  would  have  increased  x  by  unity  again,  and 
so  on  until  a  value  of  x  is  found  which  renders  f'"(x) 
positive.  When  this  value  of  x  is  found,  we  pass  to  the 
next  preceding  function.  In  this  way  we  pass  back  from 
one  function  to  another  until  we  finally  arrive  at  the 
original  function,  and  in  the  process  we  increase  the  value 
of  X  as  often  as  necessary  to  make  each  of  the  successive 
functions  positive,  and  when  the  original  function  is 
reached  and  a  value  of  x  obtained  by  this  process  which 
renders  that  original  function  positive,  it  is  unnecessary 
to  examine  the  functions  already  passed  to  see  whether 
they  are  positive  or  not,  for,  as  we  shall  presently  prove, 
the  process  insures  that  all  the  functions  passed  over  will 
be  positive  for  the  value  of  x  reached  in  this  way. 

The  important  fact  just  stated  depends  upon  this  prin- 
ciple; viz. :  if  a  rational  integral  function  of  x  and  all  its 
derivatives  are  positive  when  x^=^a^  tJieji  this  fu7iction  of  x 
is  positive  for  a7iy  value  of  x  greater  than  a. 

Let  ^{x)  be  a  rational  integral  function  of  the  rth 
degree,  and  let  its  successive  derivatives  be  represented 
by  ^'{pc),  </>''(-^),  •  •  •  ¥{j^^,  and  suppose  that  ^{x)  and  all 
its  derivatives  are  positive  when  x=a\  that  is,  ^(«), 
</)'(«),  <^"{a)  •  •  •  ^''{a)  are  all  positive. 

Now,  ^(-r+^)  =  ^(:r)  +  ^^'(;r)+|^0''(-;t:)  +  ...+.^^''(ji:) 

Hence,  ^{a^-b^  =  ^(a)-^b<^Xa)^-~^^\x 

and  since  the  coefficients  of  the  various  powers  of  b  in 
the  second  member  of  the  last  equation  are  all  positive, 
therefore  for  any  positive  value  of  b  every  term  of  the 
second  member  of  the  last  equation  is  positive;  and  hence 
f(a+b)  is  positive  whatever  positive  value  be  given  to  b. 


636  UNIVERSITY   ALGEBRA. 

Now,  in  the  process  we  have  described  in  the  special 
case  given  above,  any  one  of  the  successive  derivatives 
of /(;r)  is  itself  some  rational  integral  function  of  x,  and 
the  succeeding  derivatives  are  the  successive  derivatives 
of  this  function  of  jt,  and  hence  the  principle  just  proved 
applies,  and  as  x  is  mcreased  in  passing  back  to  previous 
functions,  therefore  it  is  unnecessary,  at  any  stage  of  the 
process,  to  go  back  and  examine  the  functions  already 
passed  over. 

KXAMPLKS. 

Find  by  each  of  the  methods  given  a  superior  limit  of 
the  positive  roots  of  each  of  the  following  equations: 

1.  x5  +  s^4_i4^3_53^2_^56^_13_0. 

2.  x'^—bx'^  —  lZx^-\-2x'^+X'-10=0, 

3.  x^—^x'^  —  lAx^  +  b^x'^+bQx+\S=0. 

4.  x'^+bx'^—lZx'^  —2x'^  +^+70=0. 

5.  x^-10x^  +  S5x'--60x+24:=0. 

6.  ;r4  +  10^^+35;t-2+50;r+24=0. 

7.  x^-Sx^-10x^  +  S0x^-Jr9x-27=0, 

8.  x^  +  Sx^-10x^-S0x'^-\-9x-\-27=0. 

9.  x^-x^-10x^-{-10x''+9x-9=0. 
10.  x^+x^  —  10x^  —  10x'^+dx-\-d=0, 

814.  Inferior  Limits.  To  find  the  inferior  limits  of  the 
positive  roots  of  an  equation,  we  transform  the  equation, 

by  substituting  —  for  x,  into  one  whose  roots  are  the  recip- 
rocals of  the  roots  of  the  proposed  equation,  then  evidently 
any  number  which  is  greater  than  the  greatest  positive  root 
of  the  transformed  equation  is  less  than  the  least  positive 
root  of  the  proposed  equation;  that  is,  a  superior  limit  of 
the  positive  roots  of  the  transformed  equation  is  an  inferior 
limit  of  the  positive  roots  of  the  proposed  equation. 


SEPARATION    OF    ROOTS.  637 

815.  Limits  of  Negative  Roots.  To  find  the  limits 
of  the  negative  roots  of  an  equation,  we  first  transform 
the  equation,  by  Art.  758,  into  one  whose  roots  are  the 
negatives  of  the  roots  of  the  proposed  equation,  then 
evidently  any  number  which  is  greater  than  the  greatest 
positive  root  of  the  transformed  equation  is,  when  its  sign 
is  changed,  negative  and  numerically  greater  than  the 
numerically  greatest  negative  root  of  the  proposed  equa- 
tion; that  is,  a  superior  limit  of  the  positive  roots  of  the 
transformed  equation  is,  when  its  sign  is  changed,  a 
superior  limit  of  the  negative  roots  of  the  proposed 
equation.  Also,  any  positive  number  which  is  less  than 
the  least  positive  root  of  the  transformed  equation  is, 
when  its  sign  is  changed,  negative  and  numerically  less 
than  the  numerically  least  negative  root  of  the  proposed 
equation;  that  is,  an  inferior  limit  of  the  positive  roots  of 
the  transformed  equation  is,  when  its  sign  is  changed,  an 
inferior  limit  of  the  negative  roots  of  the  proposed  equa- 
tion. 

KXAMPLKS. 

1.  Find  an  inferior  limit  of  the  positive  roots  of  the 
equation  x'^-—^f-x^-\-\'-x'--^-x+l=0. 

2.  Without  working  example  1,  what  would  suggest 
the  fact  that  an  inferior  limit  of  the  positive  roots  of  the 
equation  is  the  reciprocal  of  a  superior  limit  of  the  posi- 
tive roots  of  the  same  equation  ? 

3.  Find  both  superior  and  inferior  limits  of  the  nega- 
tive  roots  of  the  equation  x"^  -\-^^^-x^  +^-x'^  -\-^^^-x-{-l=0. 

4.  How  is  a  superior  limit  of  the  negative  roots  of  the 
equation  in  example  3  related  to  a  superi6r  limit  of  the 
positive  roots  of  the  equation  in  example  1  ?     Why  ? 


638  UNIVERSITY   ALGEBRA. 

5.  Show  that  the  equation  x^Sx^ +  18x'^  —  Sx—7=0 
has  a  root  between  — 1  and  0 ;  also,  a  root  between  1 
and  2;  also,  a  root  between  2  and  3;  also,  a  root  between 
4  and  5. 

6.  Show  by  Newton's  method  that  5  is  a  superior 
limit  of  the  positive  roots  of  the  equation  in  example  5. 

7.  Find  both  an  inferior  and  a  superior  limit  of  the 
negative  roots  of  the  equation 

x^  +  Sx^—5x'^  —  15x'^+4:X+12=0, 

8.  Find  both  an  inferior  and  a  superior  limit  of  the 
negative  roots  of  the  equation 

x'^+x^—bx^—bx'^+Ax+4:=0. 

816.  Thus  far  in  the  present  chapter  we  have  been 
dealing  with  the  limits  of  the  roots  of  equations.  We  now 
come  to  a  theorem  which  enables  us  easily  to  find  a  limit 
to  the  nu77iber  of  positive  roots  and  also  a  limit  to  the 
number  of  negative  roots. 

817.  Theorem.  No  /(x)  ca7i  have  7nore  positive  roots 
than  it  has  cha7iges  of  sig7is  fro77t  +  to  —  a7td  —  to  +. 

Suppose  we  have  some  polynomial  which,  when  ar- 
ranged according  to  the  descending  powers  of  x,  has  the 
signs  -f  and  —  occurring  in  the  order  given  in  M  below. 
Let  this  polynomial  be  multiplied  by  x—a,  corresponding 
to  the  introduction  of  a  positive  root  a.  Representing 
only  the  signs  in  the  multiplication,  it  is  as  follows: 

Jlf  _|_1      _|_2     _j_3     —  4     —  5     —  6     _J_7     _|_8     __9     _j_]0 

+    - 

^  _i_  1     _1_  2       13     4     5     6       17     _i    8     9     -L  1  0 

^  1     2     3       I    4       I    5     _i    6     7     8       i    9        10 

jy  _L  l        f2        fZ     4        ^5        fe       I    7        f8     9     J_  I  0     11 

We  have  written  the  signs  of  some  of  the  terms  f ,  be- 
cause a  term  having  a  +  sign  is  combined  with  a  term 


SEPARATION   OF    ROOTS.  639 

having  a  —  sign  and  it  is  unknown  which  is  numer- 
ically the  greater.  The  original  polynomial  will  be 
spoken  of  as  M,  and  the  final  product  as  P.  Figures 
have  been  attached  to  the  signs  so  that  they  may  be  spoken 
of  by  number.  It  will  be  noticed  that  while  we  use  the 
particular  order  of  signs  given  in  M,  yet  the  demonstra- 
tion we  give  is  perfectly  general,  being  applicable  whatever 
the  order  of  signs  in  the  polynomial. 

(1).  We  will  first  show  that  the  product  has  at  least 
as  vtany  changes  of  sign  as  the  Tnultiplicand, 

The  signs  in  a  are  the  same  as  those  in  M,  and  the 
signs  of  b  are  those  of  M  reversed  and  put  one  place  to 
the  right. 

Consider  any  two  consecutive  signsjof  7^,  the  (/^--l)st 
and  y^th.     They  are  either  alike  or  different. 

First,  suppose  them  alike  :^  then  the  y^th  sign  of  a  is 
the  same  as  the  >^th  sign  of  M,  while  immediately  under 
this  is  the  (/^— l)st  sign  of  b,  which  is  of  the  opposite 
kind,  so  that  the  >^th  sign  of  the  product  P  is  ?.  Second, 
suppose  the  (>^— l)st  and  y^th  signs  of  M  are  different;! 
then,  as  before,  the  >^th  sign  of  a  is  the  same  as  the  y^th 
sign  of  M,  but  the  sign  immediately  under  this  is  the 
(>^— l)st  sign  of  b,  which  is  the  (/^— l)st  sign  of  M  re- 
versed, and  hence  is  the  same  as  the  /^th  sign  of  a. 
Therefore,  the  y^th  sign  of  P  is  the  same  as  the  k\h  sign 
of  M,  unambiguous  and  unchanged.  Passing  along  the 
signs  of  M  and  P  from  left  to  right,  it  is  evident  that  the 
first  s\%r).  of  P  is  the  same  as  the  first  sign  of  y^and  that  the 
following  signs  of  P  are  all  9,  as  long  as  the  sign  in  M 
remains  the  same,  but  as  soon  as  the  sign  in  M  changes 
from  -f  to  ~  or  from  —  to  -f,  the  second  sign  of  this 
change   appears   in  P  in  the    same   position  as  in  M. 

*  lu  this  case  k  would  be  either  2,  3,  5,  6,  or  8. 

f  To  illustrate  this  reasoning,  k  may  be  taken  as  either  4, 7,  or  9  in  the  poly- 
nomial considered. 


640  UNIVERSITY    ALGEBRA. 

Therefore,  in  this  portion  of  M  and  P,  the  signs  begin 
alike  and  end  alike,  and  as  there  is  one  change  of  sign 
in  M  there  must  be  at  least  one  change  (indeed,  some 
odd  number)  in  P,  and,  as  this  holds  good  every  time  the 
sign  of  J/ changes,  therefore  there  are  certainly  as  many 
changes  of  sign  in  P  as  in  M. 

(2).  We  will  show  next  that  there  cannot  be  the  same 
number  of  changes  of  sigji  in  P  as  in  M. 

If  the  first  sign  of  the  polynomial  is  like  the  last,  the 
number  of  changes  of  sign  between  the  first  sign  and  the 
last  must  be  even\  but  if  the  first  sign  differs  from  the  last 
sign,  then  the  number  of  changes  of  sign  must  be  odd. 
Now  the  last  sign  of  P  is  necessarily  different  from  the 
last  sign  of  M.  Therefore,  if  the  first  and  last  signs 
of  i^are  alike,  the  first  and  last  signs  of  P  are  unlike  ; 
and  if  the  first  and  last  signs  of  M  are  unlike,  the 
first  and  last  signs  oi  P  are  alike.  Hence,  if  there  is  an 
even  number  of  changes  in  M,  there  is  an  odd  number  in 
P,  and  if  an  odd  number  in  M,  then  an  even  number  in  P. 
Hence  P  cannot  have  the  same  number  of  changes  as  M, 

Now  we  have  shown  :  (1.)  that  P  has  at  least  as  many 
changes  as  M\  (2.)  that  P  cannot  have  the  same  number 
as  M.  Therefore  P  has  more  changes  of  signs  than  M. 
That  is,  the  result  after  multiplying  by  x—a,  or  intro- 
ducing a  positive  root,  contains  more  changes  of  sign 
than  the  original  polynomial.  Since  at  least  one  addi- 
tional change  is  brought  in  for  each  positive  root  which 
may  be  introduced,  no  f{x)  can  have  more  positive  roots 
than  it  has  changes  of  sign  from  -f  to  — -  and  —  to  +. 

818.  Corollary.  No  f{x)  can  have  7nore  negative 
roots  than  there  are  changes  of  sig7i  from  -{-to  —  and  — 
/{?  -f ,  after  the  signs  of  all  the  odd  or  even  powers  have  been 
changed.     See  art.  758. 


SEPARATION   OF    ROOTS.  64I 

The  above  theorem  and  corollary  constitute  what  is 
known  as  Descartes'   Rule  of  Signs. 

KXAMPivKS. 

1.  Show  that  x^  —  1  has  one  positive  root  and  no  other 
real  root. 

This  function,  being  the  difference  of  like  powers,  if  divisible  by 
x  —  \',  whence  -|-1  is  a  root.  There  is  only  one  change  of  sign,  hence 
it  can  have  no  more  than  one  positive  root,  hence  none  other  than  -f- 1. 
Changing  the  signs  of  all  the  odd  or  even  powers  of  x,  there  are  no 
changes  of  signs,  hence  no  negative  roots. 

2.  Show  that  x^—1  has  two  real  roots  only,  one 
positive  and  one  negative. 

3.  Show  that  x^  +  a  has  no  real  roots 

4.  Discuss  the  roots  x"—a  when  n  is  odd,  and  also 
when  n  is  even. 

5.  Discuss  the  roots  of  x^  +  a  when  n  is  odd,  also  when 
91  is  even. 

STURM'S  THKORKM. 

819.  In  the  previous  articles  of  the  present  chapter 
we  have  found  the  limits  of  the  roots  of  equations,  and 
from  these  it  is  easy  to  separate  the  positive  roots  from' 
the  other  roots.  We  have  also  seen  by  Descartes'  rule  of" 
signs  that  the  nurnberoi  positive  roots  cannot  exceed  some- 
number  easily  found.  We  now  proceed  to  a  theorem 
which  enables  us  to  separate  from  all  the  other  roots  those 
real  roots  which  are  between  any  two  given  or  selected 
numbers,  and  also  enables  us  to  determine  the  number 
and  situation  of  the  real  roots  of  an  equation. 

820.  Notation.  Let  f{pc)  be  a  rational  integral 
function  of  x,  and  let/(:tr)  =  0  be  an  equation  with  no 
equal  roots,  and  let /'(.r)  be  the  first  derivative  oi  f{pc) 
and  let  the  operation  of  finding  the  H.  C.  F.  oifipc)  and. 


642  .  UNIVERSITY   ALGEBRA. 

f\x)  be  performed  with  the  modification  that  when  any 
remainder  is  obtained  its  sign  is  immediately  changed, 
and  let  the  process  be  continued  until  a  remainder  is 
found  which  does  not  contain  x,  and  let  this  remainder, 
like  the  others,  have  its  sign  changed.  lyCt  the  quotients 
in  succession  be  represented  ^^y  q^,  q^^  •  •  •  ^«-i>  and  the 
modified  remainders  be  represented  by  <^2(-^)>  ^sW) 
<^4(;r.),    .  .  .   ^,ix). 

The  whole  series  of  functions 

fix),     fix),       cjy.ix),    .   .  .    cl^Xx), 

we  will  call  Sturm's  Functions. 

821.  Preliminary  Propositions.  I.  T/ze  last  one  of 
Sturm's  functions  cannot  equal  zeyv.  This  is  easily  seen, 
for  <^;,(-r)  is,  by  supposition,  independent  of  x,  and  if  it 
could  equal  zero,  f{x)  and  f\x)  would  have  a  common 
measure  and  f{x)  would  have  equal  roots,  which  is  con- 
trary to  the  hypothesis. 

II.  In  Sturm's  functions,  no  two  consecutive  functions 
can  va?iish  simultaneously. 

From  the  process  described  in  Art.  820  and  the  notation 
there  used,  we  have  the  equations 

fix)  =  q^<i>^(ix)-^.^ix) 

Now,  if  any  two  consecutive  functions,  for  example: 
fix)  and  <^2(-^0  vanish  together,  the  second  equation 
shows  that  ^zi^^  ^^^^  vanishes,  and  then  <^2(-^)  and  4»'^ix) 
vanishing  together,  the  third  equation  shows  that  <t^^ix) 
also  vanishes,  and  so  on.  Therefore,  all  the  subsequent 
functions,  including  ^„ix),  vanish;  but  by  I,  0.,(a')  cannot 
vanish.  Therefore,  no  two  consecutive  fuiKnions  can 
vanish  tocrether. 


SEPARATION   OF    ROOTS.  643 

III.  When  any  function  after  fix)  vanishes^  the  two 
adjacent  functions  have  opposite  signs. 

This  is  evident  from  the  equations  tinder  II.  For 
example:  if  ^2(-^'')=^  the  third  equation  shows  that 
<^2(:r)=— <^4(x);  /.  €.,  ^'lipc)  and  <i>^{x)  have  opposite 
signs. 

822.  Sturm*s  Theorem.  If  any  two  members^  a  and 
P,  of  which  a<)S  J)e  substituted  in  turn  for  x  in  Sturfn's 
functions^  the  excess  of  the  number  of  changes  of  sign  when 
x=a  over  the  number  when  x==l3  is  equal  to  the  number  of 
real  roots  of  f(x)=0  which  lie  between  a  and  p. 

When  X  is  given  the  value  a,  Sturm's  functions,  taken 
in  order,  present  a  certain  array  of  signs  and  no  one  of 
these  functions  can  change  sign  except  when  x  passes 
through  a  value  which  makes  that  function  vanish. 

First.  lyCt  che  3,  value  of  x  which  makes  one  of  the 
functions 

/'(■^),    <^2W,    <^3W>     •    •   •    <^^X-^) 

vanish,  say  <^2('^)=^-  ^Y  II>  ^^  ^^^  preceding  article, 
neither  <^^_i(0  nor  </>^+i(0  can  equal  zero,  and  by  III, 
of  the  preceding  article,  ^r-i(^)  and  <t>r+i(c)  have  oppo- 
site signs.  Hence,  of  the  three  functions  </>^_,(<r),  <j5>^(<:), 
<^r+i  (0>  the  first  and  third  have  opposite  signs  for  a  value 
a  little  less  than  or  a  little  greater  than  c,  and  no  matter 
which  sign  cf^^Cc)  has,  the  sign  must  be  like  the  one  just 
before  it  or  just  after  it.  Therefore,  these  three  functions, 
taken  in  order,  present  one  change  of  sign  whether  the 
value  of  ;i:  be  a  little  less  or  a  little  greater  than  c;  that 
is,  the  number  of  changes  of  sign  i?i  the  whole  series  of  func- 
tions is  not  changed  at  all  by  x  passing  through  a  value 
which  makes  ^t^r^x)  vanish. 


fix+K)=Ax)^hf\x)^'^f'{x)^  .  .  .  +yJ\x)     (1) 


644  UNIVERSITY   ALGEBRA. 

Second.     Let  f  be  a  value  which  makes  f{x\  vanish; 

Now,  by  Art.  747, 

h'' ^- 

Changing  the  sign  oih,  we  have 

fi^x-K)=Ax)-hf{x)+~f"{x)+  .  .  .  ±^/«W     (2) 

Replacing  x  by  ^  and  remembering  that  f{c)—^,  we  have 

Ac-^h)=  +k/Xc)+~f'(.c-)+  ■  ■  ■  +nJ/"W     (3) 

Ac-h)=^-hf'{c)+~f"(c)-  ■  ■  ■  ±?/"C^)      (4) 

Now,  k  may  be  taken  so  small  that  in  each  of  the 
equations  (3)  and  (4),  the  second  member  has  the  same 
sign  as  its  first  term.  Hence ,  for  a  small  value  of  k ,  /(c+  Ji) 
has  the  same  sign  as  f'{c)  and  f(c—Ji)  the  opposite 
sign  from  f\c)\  i.  e,,  for  a  value  oi  x  2.  little  less  than  c, 
fix)  and  fix)  have  opposite  signs  and  for  a  value  of  x 
a  little  greater  than  c,  f{pc)  and  f'{x)  have  the  same 
signs.  Therefore,  the  number  of  changes  of  sign  in  the 
whole  series  of  Sturm^  s  functions  is  decreased  by  one  wheji 
X  passes  over  a  value  which  makes  f(jx:)=0\  i.  <?.,  when  x 
passes  over  a  real  root  of  fix). 

Now  as  the  number  of  changes  of  sign  is  neither  in- 
creased nor  decreased  except  when  x  passes  over  a  root 
of /(:r)=0  and  is  then  diminished  by  one  every  time  x 
passes  over  a  root,  therefore  as  x  changes  continuously 
from  a  to  jS  (a</3)  the  excess  of  the  number  of  changes  of 
sign  when  x=^a  over  the  number  when  x=^  is  equal  to 
the  whole  number  of  real  roots  of  /(jr) =0  between  a  and  p, 

823.  Sturm's  Theorem  for  Equal  Roots.  Leti^(jr) 
be  a  rational  integral  function  of  x^  which  we  will  sup- 


SEPARATION   OF    ROOTS.  64S 

pose  has  equal  roots,  and  let  the  first  derivative  of  F(^x) 
be  represented  by  F\x).  Apply  the  process  of  finding 
the  H.  C.  F.  of  F{x)  and  F\x)  with  the  modification 
that  every  time  a  remainder  is  found  its  sign  is  immedi- 
ately changed,  and  represent  the  modified  remainders 
by  <^2(-^)j  ^3(-^)>  ^4(-^)>  •  •  •  ^«(-^))  and  represent  the 
quotients  of  the  various  divisions  hy  q^,  q^,  q^^  *  *  *  ^«- 
Of  course  in  this  case  <^,j(^)  is  not  independent  of  Xy  but 
we  still  have  the  equations 

Fix)=q^F\x-)^^^ix) 
F'(ix')  =  q^^2(x)-^^(x) 


Let  D  represent  the  H.  C.  F.  of  F(x)  and  F'(x),  then 
the  first  of  these  equations  shows  that  D  is  also  a  factor 
of  ^2(-^)»  a^^  thence  the  second  equation  shows  that  D  is 
also  a  factor  of  ^3(^),  thence  the  third  equation  shows 
that  D  is  also  a  factor  of  ^4^(x'),  and  so  on.  Hence,  D  is 
a  factor  of  each  of  the  functions 

Fix-),  F\x),  ^2(^),  ^aW-  •  -^.W. 

Now  let  us  divide  each  of  these  expressions  by  Z>,  and 
represent  the  various  quotients  by 

wherein  </>;X^)  is  independent  of  x. 

From  the  set  of  equations  last  written  we  derive  the 
following : 

<^2W  =  ^3</>3W-^4(^) 


Now,  it  is  easy  to  see  that  these  functions  /(x),  f{x)^ 
^2W»  ^sW-  •  -^wWi  (still  called  Sturm's  functions), 


646  UNIVERSITY    ALGEBRA. 

possess  precisely  the  properties  of  Sturm*  s  functions  in 
Arts.  821  and  822,  and  hence  the  argument  of  those  articles 
applies  without  modification.  Even  the  verbal  statement 
of  Sturm's  theorem  for  the  case  of  an  equation  v^ith  equal 
roots  does  not  differ  at  all  from  the  statement  for  an  equa- 
tion without  any  equal  roots,  and  the  only  difference  in 
the  two  cases  is,  that  in  case  of  an  equation  with  equal 
roots,  Sturm's  functions  have  a  slightly  more  general 
significance  than  in  the  case  of  an  equation  without  equal 
roots. 

824.  In  the  operation  of  finding  Sturm's  functions 
we  are  at  liberty  to  multiply  or  divide  any  dividend  or 
divisor  by  any  positive  number  we  please,  for  this  does 
not  affect  the  signs  of  the  functions,  and  it  is  only  with 
the  signs  of  the  functions  that  we  are  concerned. 

To  explain  the  application  of  Sturm's  theorem  we  will 
work  out  an  example  in  detail. 

Let  us  find  the  number  and  location  of  the  real  roots 
of  the  equation 

Here  f{x)^x^-Zx''-^x-\-\Z 

/Xx)  =  3x^-6x-4: 

</>2(;r)^2;t;—5 

<t>,(x)  =  l. 
We  may  now  substitute  any  two  numbers  we  please  in 
place  of  X  in  these  functions.     Suppose  we  take  -—10  and 
-flO;  then,  arranging  the  signs  under  the  functions  to 
which  they  belong,  we  have 

/(x)    f\x)     <t>,(x)     <t>,(x-) 
when;c=  — 10    —         +  —  +      (3  changes) 

when;r=10        +         +  +  +      (no  changes) 


SEPARATION    OF    ROOTS.  647 

As  the  number  of  changes  of  sign  when  jtr— — 10  ex- 
ceeds by  three  the  number  of  changes  when  ;t:=10  we 
conclude  that  there  are  three  real  roots  of /(;r)  between 
-10  and  +10. 

Now  let  us  substitute  some  number  intermediate  be- 
tween —  10  and  +10,  say  0,  then  we  have 

Ax)     f\x)       <I>,(X)       cl>,(x) 

when  x=0         +  —  —t         +         (2  changes) 

We  now  see  that  the  number  of  changes  of  sign  when 
x= — 10  exceeds  by  one  the  number  of  changes  when 
x=0;  hence,  we  conclude  that  there  is  one  real  root  of 
/(^x)=0  between  —10  and  0.  Also  the  number  of  changes 
of  sign  when  x=0  exceeds  by  two  the  number  when 
;t:=10;  hence,  there  are  two  roots  of /(;r)=0  between  0 
and  10. 

To  find  more  nearly  the  location  of  the  negative  root, 
let  us  substitute  some  number  intermediate  between  —10 
and  0,  say  —5,  then  we  have 

when;t:=— 5     —  +  —  +  (3  changes) 

As  the  number  of  changes  when  x—  —5  exceeds  by  one 

the  number  of  changes  when  :r=0,  we  conclude  that  there 

is  one  root  between  — 5  and  0. 

Substituting  other  numbers,  we  find 

fix)     f\x)       <t>,(x)       ct>,(x) 

when;r=— 3     —  +  —  +  (3  changes) 

when:r=— 2     +  +  "~  .        +  (2  changes) 

As  the  number  of  changes  when  x=  —3  exceeds  by  one 
the  number  of  changes  when  x= — 2,  we  conclude  that 
there  is  one  root  of /(;t:)=0  between  —2  and  —3. 

As  there  is  only  one  negative  root,  we  have  located  it 
between  two  consecutive  integers,  and  hence  have  found 
its  approximate  value. 


/W 

when  x=l 

+ 

when  x=2 

+ 

when  x=S 

+ 

648  UNIVERSITY    ALGEBRA. 

Proceeding  now  to  find  approximate  values  of  the  two 
positive  roots,  we  will  substitute  different  numbers  for  x 
and  note  the  number  of  changes  of  sign  in  Sturm's 
functions. 

/XX)       CI>,(X)       c^3(^) 

—  — •  +       (2  changes) 

—  —  +       (2  changes) 
-1^           +           +        (no  changes) 

As  the  number  of  changes  of  sign  when  x=2  exceeds 
by  two  the  number  of  changes  when  x=o,  we  conclude 
that  there  are  two  roots  of  y(jr)  =  0  between  2  and  3. 

To  separate  these  two  roots  which  are  between  2  and  3, 
we  must  substitute  numbers  between  2  and  3  in  Sturm's 
functions,  and  from  the  number  of  changes  of  sign  we 
can  finally  find  numbers  between  which  these  roots  lie 
singly. 

when  x=2^       —         —  0  +  (1  change) 

Whether  we  consider  0  to  be  4-  or  —  is  immaterial, 
for  in  either  case  there  is  only  one  change  of  sign.  Hence 
as  the  number  of  changes  of  sign  when  x=2  exceeds  the 
number  of  changes  when  x=2^,  we  conclude  that  there 
is  one  root  of /(^)  =  0  between  2  and  2|,  and  as  the  num- 
ber of  changes  when  x^=2\  exceeds  by  one  the  number 
of  changes  when  x=S,  we  conclude  that  there  is  one 
root  of  y(.r)=0  between  2^  and  3. 

THEOREMS  OF  FOURIER  AND  BUD  AN. 

824.  From  a  theoretical  standpoint,  Sturm's  theorem 
leaves  nothing  to  be  desired ;  but  in  practice  the  labor  of 
obtaining  the  various  functions  called  Sturm's  functions 
is  often  so  great  as  to  discourage  the  most  patient  calcu- 
lator.  For  this  reason  the  theorem  is  not  used  much  ex* 
cept  as  a  last  resort. 


SEPARATION    OF    ROOTS.  649 

The  theorems  of  Budan  and  Fourier  are  theoretically 
much  less  perfect  than  that  of  Sturm,  but  are  so  easy  of 
application  that  one  or  the  other  of  them  is  often  used  in 
preference  to  Sturm's  theorem. 

The  theorems  of  Budan  and  Fourier  are  essentially  the 
same,  but  the  verbal  statements  as  ^iven  by  these  two 
mathematicians,  are  different.  Before  giving  these  theo- 
rems we  give  two  preliminary  propositions  to  prepare  the 
way. 

825.  Preliminary  Propositions.  I.  If  f{_x)  repre- 
sents a  rational  integral  function  of  x,  and  f^ipc)  its  first 
derivative,  a7id  if  a  is  a  root  of  f{x)  occurrijig  r  times,  then 
for  a  value  of  x  a  little  less  than  a,  f{pc)  and  f^(x)  have 
opposite  signs  and  for  a  value  of  x  a  little  greater  thaji  a, 
f{x)  and  f^ipc)  have  like  signs. 

From  Art.  747,  we  have 

Replacing  ;r  by  a  and  remembering  thaty(a)  =  0,  we  have 

A—h)- — A/iC«)+|-/2(«)  -  ■  •  •  ±?/X«) 

.•./(«-/o=(-/o/i(«)+^-/2(«)+  ■  ■■+^^y.k^) 

Similarly, 

Now,  if  a  is  a  root  occurring  in  f{x)  r  times,  /(a), 
fx^^^f'i^^  •  *  '  f-i(p)  each  vanish,  and  in  the  two  series 
just  written  the  first  terms  which  do  not  vanish  are  re- 
spectively 


6SO  UNIVERSITY    ALGEBRA. 

and  since  the  exponent  of  the  power  of  (— >^)  is  even  in 
one  case  and  odd  in  the  other,  it  is  plain  that  these  terms 
have  opposite  signs.  But  when  h  is  taken  small  enough 
each  of  the  two  second  members  above  has  the  same  sign 
as  the  first  term  which  does  not  vanish.  Therefore, 
f(a—Ji)  and/i(a— /2)  have  opposite  signs  for  some  small 
value  of  h. 

If,  now,  throughout  the  discussion  just  given,  the  sign 
of  h  be  changed,  then  the  first  terms  in  the  two  series 
which  do  not  vanish  have  like  signs,  and  therefore 
/(cL-^-Ii)  and  /i(a  +  /2)  have  like  signs. 

II.  If  a  is  a  root  occurring  iji  f(^x)  r  times  then  for  a 
value  of  X  a  little  less  thaii  a  the  series  of  functions  f{x), 
fx(x),  '  .  •  fr(x:),  have  signs  alternately  +  a7id  — ,  and  for 
a  value  of  x  a  little  greater  than  a  each  of  these  fu7ictio7is 
has  the  same  sign  as  fix). 

From  I,  f{x^  andyj(x)  have  opposite  signs  for  a 
value  of  ;t:  a  little  less  than  a.  Also,  as  f^i^x')  is  the 
derivative  of/i(jr)  the  argument  of  I  shows  that  f^ix) 
and  f<i{x)  have  opposite  signs  for  a  value  of  x  a  little 
less  than  a. 

The  same  statement  applies  to/2(:r)  and  f^(x),  and 
so  on.  Hence,  in  the  series  of  functions  f{x),  f\{pc), 
fii.^)^'  •  ' fr{.^)>  ^^ch  function  has  the  opposite  sign 
from  the  preceding  function;  that  is,  the  signs  of  these 
functions  are  alternately  +  and  —  for  a  value  of  ;r  a  little 
less  than  a.  Again,  the  argument  of  I  applies  to  any  two 
consecutive  functions  of  the  series/(:r),/i(;r),/2(^),  •  •  • 
fr(x)  when  x  has  a  value  a  little  greater  than  a,  and  shows 
that  any  two  consecutive  functions  of  this  series  have 
like  signs;  hence,  for  a  value  of  jr  a  little  greater  than  a 
all  these  functions  have  like  si^ns.  But  since  a  is  not  a 
root  of /^(;r)  =  0,  it  follows  that  fr{x)  does  not  change  its 


SEPARATION   OF   ROOTS.  651 

sign  when  x  passes  through  the  value  a.     Hence,  when  x 
is  a  little  greater  than  a,  each  of  the  functions 

/W)    /iW,    /2W,  •  •  •  /.W 
has  the  same  sign  as  fripc). 

826.  Fourier's  Theorem.  If  two  numbers,  a  and  P, 
(a<y8)  de  substituted  for  x  in  the  series  of  functions 

fix),Mx),Mx),...f,lx-), 
then  the  number  of  real  roots  of  f(^x')=0  which  lie  between 
a  and  (3  is  ?iot  greater  than  the  excess  of  the  7iumber  of 
changes  of  sign  in  this  series  of  functions  when  x=a  over 
the  7iumber  of  changes  when  x=l3  and  when  the  number 
of  fvots  is  7iot  equal  to  this  excess  it  is  less  by  some  even 
number. 

Let  us  suppose  x  to  start  with  the  value  a  and  increase 
continuously  to  the  value  ^.  Then,  in  this  increase,  we 
have  the  following  four  cases  to  consider : 

First,  X  may  pass  through  a  value  which  is  a  root 
which  occurs  but  once  in  f{x)=^. 

Second,  x  may  pass  through  a  value  which  is  a  root 
which  occurs  r  times  in  /(jt:)  =  0. 

Third,  x  may  pass  through  a  value  which  is  a  root 
of  one  of  the  derived  functions,  say  fix),  which  root  oc- 
curs but  once  in/X^)  =  0  and  does  not  occur  at  all  in 
either  /,_i(^)=0  or  f^^{x)=0. 

Fourth,  X  may  pass  through  a  value  which  is  a  root 
occurring  r  times  in  /^(^)=0,  and  not  occurring  at  all  in 
fs-,(,x). 

These  four  cases  we  will  consider  in  order. 

First  case.  Let  c  be  the  single  root  of /(:r)=0.  Then, 
from  Art.  825,  I,  fix)  and  /^  (x)  have  opposite  signs  for 
a  value  of  Jtr  a  little  less  than  c,  and  like  signs  for  a  value 
of  jt:  a  little  greater  than  c.     Therefore,  in  the  series  of 

functions 

fix),    f^ix),    f^ix),  •  .  . ,  fix), 


652  UNIVERSITY    ALGEBRA. 

there  is  one  change  of  sign  lost  as  x  passes  through  the 
value  c. 

Seco7id  case.  lyet  c  be  the  root  occurring  r  times  in 
/(jr)=0.    Then,  from  Art.  825,  II,  the  series  of  functions 

have  signs  alternately  +  and  —  for  a  value  of  ^  a  little 
less  than  c,  and  all  have  like  signs  for  a  value  oi  X2i  little 
greater  than  c.     Therefore  in  the  series  of  functions 

there  are  r  changes  of  sign  lost  as  x  passes  through  the 
value  c. 

Third  case.  I^et  c be  vhe  single  root  of  fsipc)  =  0.  Then, 
by  Art.  825,  I,  fsipc)  and  yS+i(^)  have  opposite  signs  for 
a  value  of  x  a  little  less  than  c,  and  like  signs  for  a  value 
of  X  a  little  greater  than  c.  Now,  when  x  has  a  value 
either  equal  to  ^  or  a  little  greater  or  a  little  less  than  c, 
the  sign  of /s-i(x)  is  either  the  same  as  or  opposite  to 
that  of  /s+ 1  (:r).  First,  suppose/^. ^  (x)  has  the  same  sign 
as  /s+i(^')-  Then,  since  the  sign  of /^(;r)  is  at  first  oj/^iJ'^- 
site  to  and  then  the  same  as  that  of/s+i(x),  it  folla^s 
that  in  the  three  functions  y]._i(jir),  fs(x),  yS+i(:r)  th^^r^ 
are  two  changes  of  sign  lost  as  x  passes  through  c ;  via.  • 
one  between  yS(^)  andyS+xW  ^^^  another  between  7^_i  {x) 
and/^(;t:).  Second,  suppose  /x_i(^)  and  /^+i(^)  have 
opposite  signs.  Then,  although  fs{x)  changes  its  sign  a? 
X  passes  through  c,  still  there  is  neither  gain  nor  loss  in 
the  number  of  changes  of  sign,  for  in  any  case  the  sign 
of  y^(;»;)  is  the  same  as  that  of  one  and  opposite  to  that  of 
the  other  of  the  two  adjacent  functions. 

On  the  whole,  therefore,  as  x  passes  through  a  single 
root  of  fs(,x)  =  0  one  of  two  things  must  happen ;  either 
there  are  two  changes  of  sign  lost  or  the  number  of 
changes  is  not  affected  at  all. 


SEPARATION   OF    ROOTS.  653 

Fourth  case.  In  this  case  a  root  occurs  r  times  in 
y^(^)==0.  Let  us  first  consider  r  an  even  number  and  let 
c  represent  a  root  which  occurs  r  times  in  yS(;r)=0.  By 
Art.  825,  II,  the  series  of  functions 

/.W>       Z.+  lW,   •    •    •     f stripe) 

have  signs  alternately  +  and  —  when  :t:  is  a  little  less 
than  c  and  all  have  the  same  sign  when  Jt:  is  a  little  greater 
than  c,  and  this  sign  is  the  same  as  the  sign  of  yS+;,(;»;). 
Now,  for  a  value  of  a  little  less  than  c, 

the  sign  ofyS+^_i(;tr)  is  opposite  to  the  sign  of  j^+^(;tr) 
.*.  the  sign  ofyS+^_2(-^)  is  the  same  as  the  sign  of/i+^(jt:) 
.'.  the  sign  of  j^^.^. 3 (:r)  is  opposite  to  the  sign  ofy^+^(:r) 
and  so  on.  Hence,  since  r  is  an  eve7i  number  the  sign  of 
fsipc)  is  the  same  as  the  sign  of /^^.^.(jr). 

Now,  X  still  being  a  little  less  than  c,  the  sign  oifs-^{pc) 
may  be  the  same  as  the  sign  of /^^^(jt),  and  hence  the 
same  as  the  sign  of /X-^)>  or  it  may  be  opposite  to  the 
sign  oi  fs^r^P^^i  and  hence  opposite  to  the  sign  oi  fsKx), 

First,  if  the  sign  oi  fs-^ipc)  is  the  same  as  the  sign  of 
fsix)  there  is  no  change  of  sign  between  y^_i(^)  and/^(:i:) 
when  ;r  is  a  little  less  than  c.  Therefore,  by  Art.  825,  II, 
when  ^  is  a  little  less  than  c,  the  series  of  functions 

fs-^{x),    fix),    /.+  i(-r),-  •  •  /.+.(-:r) 
present  r  changes  of  sign,  and,  since  when  :r  is  a  little 
greater  than  c,  this  same  series  of  functions  all  have  like 
signs,  therefore,  as  ^  passes  through  c,  there  are  r  changes 
of  sign  lost. 

Second,  if,  when  :r  is  a  little  less  than  c,  the  sign  oif,_^  {x) 
is  opposite  to  the  sign  of  yS(^),  then  there  is  one  change 
of  sign  between  fs-^{pc)  and  fs{pc),  and  therefore  when  x 
is  a  little  less  than  c,  the  series  of  functions 

fs-xix).    fsipc).    /.+  i(-^),  •  •  •  /.+.(-^) 
present  r+ 1  changes  of  sign,  and  since  when  ;t:  is  a  little 


654  UNIVERSITY   ALGEBRA. 

greater  than  c  this  series  of  functions,  with  the  exception 
ofyS_i(-^),  all  have  like  signs,  therefore  the  series  of 
functions 

/.-iW  fs{pc).  Z+iW,  •  •  •  /.+.(-^0 
present  one  change  of  sign  when  ;r  is  a  little  greater  than  c. 
Therefore,  as  x  passes  through  c  there  are  r  changes  of 
sign  lost.  Hence,  we  conclude  that  whether  the  sign  of 
y^_i(^)  is  the  same  as  or  opposite  to  the  sign  of /i_^^(;f) 
there  are  r  changes  of  sign  lost  in  the  series  of  functions 

as  X  passes  through  the  value  c. 

It  is  true  that  these  are  only  part  of  the  whole  series  of 
functions  we  are  dealing  with  in  Fourier's  theorem,  but 
of  the  whole  series  from  f{pc)  to  fnix)  the  only  ones  that 
change  sign  when  x  passes  through  c  are  some  of  those 
from/X-^)  \.o  fsJ^r-^ix)  inclusive.  Therefore,  r  being  an 
eyen  number,  as  x  passes  through  c  there  are  r  changes 
of  sign  lost  in  the  whole  series  of  functions 

Let  us  next  take  the  case  in  which  r  is  an  odd  number. 
As  before, 

the  sign  of/,.+^_i(:t:)  is  opposite  to  the  sign  of/^4.^(x) 
.*.  the  sign  of/^+^_2(^)  is  the  same  as  the  sign  of/^+^(x) 
.-.  the  sign  oi  fsJ^r-zipc)  is  opposite  to  the  sign  of/^+;.(;t'), 
and  so  on.     Hence,  since  r  is  an  odd  number,  the  sign 
oifsipc)  is  opposite  to  the  sign  of/^+^(x). 

Now,  X  still  being  a  little  less  than  c,  the  sign  oifs-^  {x) 
may  be  the  same  as  the  sign  oif stripe)  and  hence  opposite 
to  the  sign  oi  fs{pc),  or  it  may  be  opposite  to  the  sign  of 
fs\r{x^,  and  hence  the  same  as  the  sign  oifs{pc). 

First,  if  the  sign  of  yS_i(:r)  is  the  same  as  the  sign  of 
fs^r^pc)  and  hence  opposite  to  the  sign  oi  fXx)  there  is 


SEPARATION    OF    ROOTS.  '  655 

one  change  of  sign  between  yS-i(-^)  and  fs^x)  when  x  is 
a  little  less  than  c.  Therefore,  by  Art.  825,  II,  the 
series  of  functions 

present  r+l  changes  of  sign,  and,  since  when  ;t:  is  a  little 
greater  than  c  this  same  series  of  functions  all  have  like 
signs,  therefore,  as  x  passes  through  c  there  are  r+1 
changes  of  sign  lost. 

Secojid,  if  the  sign  oi  fs-^^pc)  is  opposite  to  the  sign  of 
f{pc)  and  hence  the  same  as  the  sign  of //.r),  there  is  no 
change  of  sign  between /^_i(;t:)  and /X-^)  when  ;r  is  a 
little  less  than  c.  Therefore,  by  Art.  825,  II,  the  series 
of  functions 

fs-xipC).       fsix).       fs4-l{x),-    .    .     fs+r(x) 

present  r  changes  of  sign  when  x  is  a  little  less  than  c. 
Also,  by  Art.  825,  II,  this  same  series  of  functions,  with 
the  exception  of  fs-\{x),  all  have  like  signs  w^hen  ;i:  is  a 
little  greater  than  c.     Therefore,  this  series  of  functions 

present  one  change  of  sign  when  Jt:  is  a  little  greater  than  c. 
Therefore,  as  x  passes  through  c  there  are  r—1  changes 
of  sign  lost.  Hence,  when  r  is  an  odd  number,  there  are 
either  r+1  or  r—\  changes  of  sign  lost  as  x  passes 
through  the  value  c.  But  when  r  is  an  odd  number  r+1 
and  r—l  are  both  even  numbers.  Thus,  we  see  that  in 
this  fourth  case  whether  r  is  an  even  or  an  odd  number 
there  is  always  an  even  number  of  changes  of  sign  lost 
as  X  passes  through  the  value  c. 

Reviewing  now  the  four  cases  of  the  demonstration 
just  given,  we  conclude 

I.  That  as  x  increases  from  a  to  /?  there  is  never  a  gain 
in  the  number  of  changes  of  sign. 

II.  That  each  time  x  passes  through  a  single  root  of 
y*(;i;)=0  one  change  of  sign  is  lost. 


6S6  UNIVERSITY   ALGEBRA. 

III.  That  each  time  x  passes  through  a  root  which 
occurs  r  times  in  f(x')  =  0,  r  changes  of  sign  are  lost. 

IV.  That  in  no  case  except  when  x  passes  through  a 
root  of /(jr)  =  0  can  an  odd  number  of  changes  of  sign  be 
lost.  Hence,  from  these  results  it  follows  that  the  whole 
number  of  changes  lost  during  the  change  in  x  from  a  to 
p  must  be  equal  to  the  number  of  real  roots  of  /(.r)=0 
between  a  and  yS,  or  must  exceed  the  number  of  these 
roots  by  some  eveti  number. 

827.  Budan's  Theorem.  If  the  roots  of  the  equation 
f(x')=Q  be  diminished  first  by  a  a?zd  then  by  )8(a<yS),  the 
number  of  7'eal  roots  of  f(x)  =  0  between  a  and  /3  cannot  be 
greater  than  the  excess  of  the  number  of  changes  of  sign  in 
the  first  tra7isformed  equation  over  the  number  of  cha7iges 
in  the  second  transforfued  equation. 

This  theorem  is  easily  seen  to  be  included  in  Fourier's 
theorem,  for,  by  Art.  761,  the  two  transformed  equations 
are 

/(-)+/l(-)>'+/2(«)|^+  •    •   •  +/<(«)£=0- 

and  from  the  preceding  article  the  truth  of  this  theorem 
is  now  evident.  Budan's  statement  of  the  theorem  is 
rather  easier  to  apply  to  numerical  equations  than  Fourier's 
because  the  transformation  of  the  given  equation  is  so 
easily  accomplished  by  the  method  of  Art.  762. 

828.  It  is  well  to  notice  that  Fourier's  theorem  really 
includes  both  Descartes'  rule  of  signs  and  Newton's 
method  of  finding  the  limits  of  the  roots  of  equations. 


SEPARATION   OF    ROOTS.  657 

Descartes^  rule  of  signs.     If  we  take 

ioxf{pc),  then  we  have  the  following  identities: 

f^(x^  =  na^x^-'-^(n-l)a,x^-'^  +  (n-2')a^x^-^  +  ,..+a„_^ 
f^(x)  =  n(n—l)aQX''-^  +  (n—l){n—2)a^x''-^-^ h«„-2 

From  these  equations  it  is  easily  seen  that  when  x=0 
the  series  of  functions 

reduce  to  the  coefficients  in  /(x)  taken  in  reverse  order. 
But  when  x  is  taken  equal  to  some  sufficiently  large 
number,  say  d,  all  these  functions  have  the  same  sign 
and  are  all  positive  or  negative  according  as  a  q  is  positive 
or  negative. 

Therefore,  the  number  of  changes  of  sign  in  the  coeffi- 
cients of  /(x)  is  equal  to  the  excess  of  the  number  of 
changes  of  sign  in  the  functions 

when  x=0  over  the  number  of  changes  when  x=d. 
Hence,  by  Fourier's  theorem,  the  number  of  real  roots 
of /(;i:)=0  between  0  and  d  is  not  greater  than  the  num- 
ber of  changes  of  sign  in  /(-^).  But  d  is  supposed  to  be 
a  number  greater  than  the  greatest  positive  root  of/(x)=0, 
therefore  the  number  of  roots  between  0  and  d  is  the 
number  of  positive  roots  of  the  equation.  Hence,  the 
number  of  positive  roots  of  /(x)=0  is  not  greater  than 
the  number  of  changes  of  sign  in  /{x)y  and  this  statement 
is  Descartes*  rule  of  signs. 

42- U.  A. 


658  UNIVERSITY    ALGEBRA. 

Newton^  s  method  of  finding  limits  of  roots.  Suppose  h 
is  some  number  which,  substituted  for  x,  renders  each  of 
the  functions 

positive.  But,  as  was  shown  in  Art.  813,  each  of  these 
functions  is  also  positive  when  x=h-\-k.  Therefore,  it 
follows  by  Fourier's  theorem  that  there  are  no  roots  be- 
tween h  and  ^+>^  however  great  k  may  be.  Therefore,  ^  is  a 
superior  limit  of  the  positive  roots  of  the  equation  y(jtr)=0. 

KXAMPI^KS. 

1.  Show  by  Sturm's  theorem  that  the  equation 
;t:^  +  6:1;^  +  10;r— 1=0  has  only  one  real  root  and  that  this 
root  is  less  than  unity. 

2.  Apply  Sturm's  theorem  to  the  equation 

3.  Apply  Sturm's  theorem  to  the  equation 

.r4  +  2;t;2— 4;^-fl0=0. 

4.  Apply  Sudan's  theorem  to  the  equation 

;t:4-f3;i;3+7^2_j.i0;r+l-=0. 

[5.     Show  that  the  equation 

;^;5__3^4_24;r3+95;t;2_4e^_101=0. 
has  all  its  real  roots  between  —10  and  10. 

6.  Find  the  integral  part  of  the  positive  root  of  the 
equation  ji:^— 4;>;— 12=0. 

7.  Show  by  means  of  Sturm's  theorem  that  the  equa- 
tion ji:^  +  ll;i;^  — 102;i;+181=0  has  two  roots  between  3 
and  4. 

8.  Apply  Sturm's  theorem  for  equal  roots  to  the  equa- 
tion ^*-5;<;3+9;j;2_7^_|.2=0. 


CHAPTER  XXXV. 

NUMKRICAIy  EQUATIONS. 

829.  Although  a  general  solution  of  the  general 
equation  of  the  nth  degree  does  not  exist,  yet  it  is  possi- 
ble to  find  the  real  roots  of  an  equation  of  any  degree, 
provided  the  given  equation  has  numerical  coefficients. 
The  process  of  solution  depends  upon  the  properties  of 
f{pc)  already  established,  and  is  satisfactory  in  all  respects, 
giving  the  value  of  the  roots  exactly,  if  commensurable, 
or  to  any  desired  degree  of  approximation  if  incommen- 
surable. 

830.  We  begin  with  an  illustration  of  the  general 
method  by  solving  a  particular  example,  and  shall  after 
wards  summarize  the  process  in  a  general  statement  of 
advice  for  any  case.  Suppose  it  is  required  to  solve  the 
equation 

I.  Put  x-=\y,  which  transforms  the  given  equation 
into  the  following  (Art.  760),  which  has  no  fractional 
roots  (Art.  759). 

<^(_y)=j,5_3j/4_i6^3_^28j/2+72j/+32=0.         (2) 

The  real  roots  of  this  must  be  either  whole  numbers  or 
incommensurable  numbers. 

II.  In  seeking  for  integral  roots,  we  need  only  search 
among  the  factors  of  the  absolute  term  (Art.  757),  which 
are:  zhl,  ±2,  ±4,  ±8,  ±16,  ±32.     We  may  now  test 


66o  UNIVERSITY    ALGEBRA. 

these  by  dividing  (2)  by  y  minus  each  of  them  (Art.  741). 

Thus: 

1     -3     -16     +28     +72     +32     (+1 

+  1-2     -18     +10     +82 

1     -2     -18     +10     +82  +114 

+  1  is  not  a  root. 

1     -3     _16     +28     +72  +32     (+2 

+  2-2     -36     -16  +112 

1     -I     -.18     -  8     +56  +144 
+2  is  not  a  root. 

1     _3     _16     +28     +72  +32     (+4 

+  4+4     -^48     -80  -32 
1     +1     _12     -20     -  8  0 

+4  /^  a  root. 

This  last  quotient  gives  us  an  equation  • 

y  +_^3  _i2y2  _90y~8=0.  (3) 

which  is  of  one  lower  degree  than  (1)  and  which  contains 
the  remaining  roots. 

Now  we  need  try  only  factors  of  8,  of  which  we  have 
already  tried  +1,  +2,  +4.  By  trial  it  is  found  that  +8 
is  not  a  root.  In  general  it  is  better  to  test  the  small 
negative  factors  before  testing  the  large  positive  factors. 
Dividing  by  J/+1,  corresponding  to  a  root  —1,  we  have 
1  +1  _12  -20  +8  (-1 
-1      ^  0     +12     -8 

1        0     -12     -  8        0  —1  is  a  root. 

The  equation  containing  the  remaining  roots  is 

^3_i2y-8=0.  (4) 

Using  the  untried  factors  of  8,  we  find  no  more  integral 
roots.  Hence,  the  equation  must  have  three  incommen- 
surable roots,  or  one  incommensurable  and  two  imaginary. 


NUMERICAL  EQUATIONS.  66l 

III.  It  is  now  well  to  test  for  eqtial  roots,  for  if  two  or 
three  of  these  incommensurable  roots  are  equal,  we  can 
find  them  very  readily  by  Art.  769.  But  y^  —  Vly—Z 
and  the  derivative  Zy'^—Vl  have  no  common  divisor, 
hence  there  are  no  equal  roots. 

IV.  We  next  attempt  to  locate  the  incommensurable 
roots  by  assigning  different  values  to  y  and  calculating 
the  corresponding  values  of  <^(  jj')  (Art.  808).  If  we  put 
any  value,  a,  for  y\n  <^(jk),  we  can  compute  the  value  of 
<i>{ct)  by  the  short  method  of  division  ;  for  ^{a)  is  the  re- 
mainder when  ^(y^  is  divided  by  j/— ^.     (Art.  739). 

Thus,  from  </)(jj/)=jk^+0j/2 -12y—8=0,  we  get 

1  +0  -12  -8  (+2      1  +0  -12  -8  (+3 
+  2  +4  -16  +3  +9  -9 


1  +2  -8  -24=i?=(^C2)  1  +3  -3  -17=ie=(^(3) 

1  +0  -12  -8  (+4     1  +0  -12  -  8  (-2 
+  4  +16  4-16  -2  +  4  +16 


1  +4  +  4  +  8=7?=c^(4)  1-2-8  +8=i?=(?f>(-2) 

etc.,    etc.,    etc.,    etc. 

In  like  manner  we  find  the  values  of  <^(jy)  when  y  has 
other  values  and  tabulate  them  as  in  the  mar- 


gin.    The   first   column   contains  the  values  JL. 


assigned  to  y  and  the  second  contains  the  cor-  ~^ 
responding  values  of  ^{^y).  It  is  then  noticed  _^ 
that  there  is  a  root  between  +3  and  +4  and  —  i 
between  0  and  —1  and  between  —3  and  —4.  0 
(Art.  808\  +1 

V.     The   roots   thus  located  can  be  deter-  J"^ 
mined  to  any  degree  of  accuracy  in  the  manner    ,  ^ 
following.     Since  equation  (4)  has  a  root  be- 
tween +  3  and  +4,  the  first  figure  of  the  root  is  3.  Then 


<^(J') 


-21 
+  1 
+  8 
+  3 
-  8 
-17 
-24 
-17 
+  8 


662 


UNIVERSITY   ALGEBRA. 


transform  the  equation,  by  Horner's  method,  into   one 
whose  roots  are  3  less.     The  work  is  as  follows : 
1     +0     -12     -8     (3 
+  3     +9-9 


-1-3-3 
+  3     +18 


+6 

+  3 


+  15 


-17 


(5) 


1  I  +9 
The  resulting  equation  is 

j)/8+9j/2+15j/-17=0. 
This  must  have  a  root  between  some  of  these  values  : 
.0,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9,  1.0.     By  trial  it  is  found 
to  lie  between  .7  and  .8,  thus  : 

1     +9.0     +15.00     -17.000    (.7 
+  .7     +  6.79     +15.253 


1     +9.7     +21.79 


1.747 


+9.0     +15.00     -17.000 
+  .8     +  7.84     +18.272 


(.8 


1     +9.8     +22.84     +  1.272 
The  first  figure  of  this  root  of  (5)  is  therefore  7,  which 
is  the  second  figure  of  a  root  of  (4). 

Now  depress  the  roots  of  (5)  by  .7,  by  Horner's  method. 
The  work  is  as  follows : 

1     +9.0    +15.00    -17.000    (.7 
+  .7     +  6.79     +15.253 


+9.7 
+  ,7 


+21.79 
+  7.28 


1  +10.4 

.T_ 

1|+11.1 


+29.07 


-  1.747 


NUMERICAL   QUATIONS. 


663 


The  resulting  equation  is 

jj/3  +  ll.iy+29.07>/-l. 747=0.  (6) 

This  must  have  a  root  between  some  of  these  values  : 
.00,  .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .10.  By  trial 
it  is  found  to  lie  between  .05  and  .06,  thus  : 

1     +11.10     +29.0700     -1.747000      (.05 
+  .05         +.5575     +1.481375 


1     +11.15 

+  29.6275 

-  .265625 

1     +11.10 

+  29.0700 

-1.747000 

(.06 

+  .06 

+  .6696 

+  1.784376 

1     +11.16     +29.7396     +  .037376 

Hence,  5  is  the  first  figure  of  a  root  of  (6),  which  is  the 
second  figure  of  a  root  of  (5),  which  is  the  third  figure 
of  a  root  of  {^))  therefore,  this  root  of  (4),  correct  to  three 
figures,  is  3.75. 

We  now  depress  the  roots  of  (6),  by  .05.     The  work 
is  as  follows : 

1     +11.10     +29.0700     -1.747000 
.05  .5575        1.481375 


(.05 


1 


+  11.15 
.05 


+29.6275 
.5600 


-  .265625 


1     +11.20 
.05 


+30.1875 


1  I  +11.25 
The  resulting  equation  is 

jj/3  4.ii.25y+30.1875y-.2656250=a  (7) 

By  trial  this  equation  is  found  to  have  a  root  between 
.008  and  .009.  Hence  8  is  the  first  figure  of  a  root  of  (7), 
the  second  figure  of  a  root  of  (6),  the  third  figure  of  a  root 
of  (5),  or  the  fourth  figure  of  a  root  of  (4).    Hence  a  root 


664  UNIVERSITY   ALGEBRA. 

of  (4),  complete  to  four  figures,  is  3.758.  It  is  evident 
that  a  root  cau  be  determined  to  any  degree  of  accuracy 
by  a  continuation  of  this  process. 

In  like  manner  we  could  find  the  other  incommen- 
surable roots  of  (4),  either  the  one  between  —8  and  —4, 
or  0  and  —1.  For,  an  incommensurable  negative  root 
can  be  found  if  the  negative  roots  be  transforvted  into  posi- 
tive ones  before  apply vig  Horner' s  Method.  This  can  be 
done  by  Art.  758. 

We  may  obtain  the  approximate  values  of  the  other 
two  roots  from  the  quadratic  resulting  from  removing  the 
approximate  root  from  (7). 

1     +11.250     +30.187500     -.265625000     (.008 

.008  .090064        .242220512 

1     +11.258     +30.277564 

j/2  +  ii.258j/+30.277564=0  (8) 

y +  11. 258JI/+31. 685641  =  1.408077 
whence,  jj/=— 4.443  and  —6.815. 

But,  remembering  that  these  roots  are  3.75  less  than  those 
of  (4),  we  really  have,  as  the  roots  of  (4), 
j/=-.693and  -3.065. 
VI.     Finally,  collecting  all  the  values  of  y  found,  and 
remembering  that  x=^\y,  we  obtain  these  results  as  the 
solution  of 

j^=-4,  or  -1,  or  +3.758  +  ,  or  -.693,  or  -3.065 
^=  -2,  or  -^,  or  +1.879  ...  or  -.345  ...  or  —1.533  ... 

831.  The  Principle  of  Trial  Divisors.  The  pro- 
cess given  above  for  finding  the  successive  figures  of  a 
root  would  be  found  in  practice  to  be  very  tedious,  since 
each  successive  figure  is  determined  from  among  several 


NUMERICAL    EQUATIONS.  665 

digits  by  actual  trial.     But  it  will  be  found  that  after  one 
or  two  figures  of  the  root  are  obtained  as  above,  that 
a  suggestion  of  the  next  figure  can  be  obtained  in  a  very  * 
simple  way.     To  illustrate  this,  consider  equation  (7)  in 
the  last  article: 

j/3  +  11.25y  +  30.1875j/~.^65o25=0. 
We  know  that  y  is  some  number  of  thousandths  plus 
something.  We  are  determining  the  figures  of  the  root 
one  at  a  time,  and  at  present  merely  desire  the  number  of 
thousandths.  By  proper  transformations  in  the  equation 
it  is  evident  that 

_  .265625 

-^"j/'  +  11.25j/+30.1875* 
Now,  since  jK  is  known  to  be  a  fraction,  and  less  than  one 
hundredth,  the  most  valuable  term  in  the  denominator  of 
the  fraction  is  30.1875;  for,  the  higher  the  powers  oi  y 
the  less  account  they  are  when  _y  is  a  fraction.     Hence 

-^^sral  ^ea^ly=-000S79+  (9) 

Whence,  it  is  quite  certain  that  8  is  the  first  figure  of  y, 
or  the  fourth  figure  of  a  root  of  (4).  Since  this  same 
reasoning  would  apply  whatever  the  given  equation  might 
happen  to  be,  therefore,  in  any  case,  when  two  or  three 
figures  of  a  root  have  been  obtained,  a  suggestion  of  the 
next  figure  can  be  had  by  dividing  the  absolute  term  of 
the  depressed  equation  by  the  coefficient  of  the  first  power 
of  the  unknown  number. 

This  is  known  as  the  Principle  of  Trial  Divisors. 
Of  course,  as  we  find  more  figures  of  the  root  and  contin- 
ually depress  the  roots  of  the  equation,  the  smaller  y 
becomes  and  the  more  surely  can  we  rely  upon  the  sug- 
gested value.  Thus,  two  figures  of  the  approximate 
value  of  y,  given  by  (9),  are  really  correct,  as  will  be  seen. 


666 


UNIVERSITY    ALGEBRA. 


832.  Arrangement  of  Work.  We  now  give  a  more 
elaborate  arrangement  of  the  work  of  Horner's  method 
as  used  in  article  830.  The  lines  that  extend  across 
the  page  are  the  divisions  between  the  successive  depres- 
sions. Follow  each  line  accross  and  the  numbers  beneath 
it  are  the  coefficients  of  the  equation  with  roots  depressed. 
The  fifth  figure  of  the  root  was  obtained  by  the  method 
of  trial  divisor;  that  is,  by  dividing  .023404488  by 
30.367756. 


11.274 


-12 

_9 

-  3 
i8 


15.00 

6.97 


-  8 

-  9 


(3.7589 


-17.000 

15-253 


1.747000 

1.481375 


9.0 

21.79 

-  .265625000 

.7 

7.28 

.242220512 

9.0 

29.0700 

-  .02uoms 

.7 
10.4 

•7 

.5575 

29.6275 

.5600 

11.10 
•05 

30.187500 

. 090064 

II. 15 

30.277564 

•05 

.090192 

11.20 

30.367756 

•  05 

11.250 

.008 

11.258 
.008 

» 

11.266 

.008 

833.  Summary.  The  equation  just  solved  illustrates 
the  method  of  procedure  by  which  we  may  determine  the 
real  roots  of  any  numerical  equation.  We  briefly  sum- 
marize the  process  in  the  following  statement  of  advice 
for  any  case: 


NUMERICAL   EQUATIONS.  ^6j 

Any  numerical  equation  being  given : 

I.  Transform  the  equation,  if  necessary,  so  that  the 
coefficient  of  the  highest  power  in  f{pc)  shall  be  unity  and 
none  of  the  other  coefficients  shall  be  fractions. 

II.  Search  among  the  positive  and  negative  factors  of 
the  absolute  term  for  integral  roots  by  dividing  f{x)  by 
X  minus  each  factor  by  synthetic  division.  Use  the  numeri- 
cally smallest  values  first.  Depress  the  degree  of  the 
equation  whenever  a  root  is  found. 

III.  When  all  integral  roots  are  found,  test  for  equal 
roots,  by  noting  whether  the  function  and  the  first  deriva- 
tive have  a  common  divisor. 

IV.  Tabulate  the  function  and  locate  the  incommen- 
surable roots  by  Art.  808,  and  approximate  them  as 
desired  by  Horner's  method. 

V.  As  soon  as  a  quadratic  equation  is  obtained,  solve 
it  in  the  ordinary  way. 

834,  The  separation  of  the  roots  may  be  effected  by 
Sturm's  theorem,  as  explained  in  Art.  824.  But  the 
labor  of  computing  Sturm's  functions,  renders  the  appli- 
cation of  that  theorem  undesirable  save  in  exceptional 
cases. 

Among  these  exceptional  cases  may  be  mentioned  the 
one  in  which  an  equation  has  two  or  more  roots  lying 
between  consecutive  integers.  For,  if  there  be  an  even 
number  of  such  roots,  they  would  not  be  pointed  out  by 
Art.  808,  but  Sturm's  theorem  would  indicate  their 
presence.     Thus,  see  the  example  in  Art.  824. 

KXAMPI.KS. 

Solve  the  following  equations  : 

1.  j»;3— 9;r2+23;r— 15=0. 

2.  .;»;4_|.2;^3_2l;t:2_22;,;-f40=0. 


66S 


UNIVERSITY   ALGEBRA. 


3.  :tr^--5jir3  — 13:^2 +52jr  4- 60=0. 

5.  x^—x^—x^  +  19;i:— 42=0. 

6.  Jtr^— 3jtr4— 9:^3 -f21;i;2-10;r+ 24=0. 

7.  ;«;3--4;tr— 12=0. 

8.  ;t:3— 24;i;+44=0. 

9.  ^3^10;t:2+6ji;— 120=0. 

10.  ;r3+:r— 3=0. 

11.  ;i;3-3;i;+l=0. 

12.  3;i;3  4-5;»r-40=0. 

13.  x3-17=0. 

14.  ;i;4-15=0. 

15.  x^—3.5x^--h2x-}-2=0. 

16.  .r3-15^-5=0. 

17.  ;t:4-8-r^  + 12.^2  _^3;t:--4=0. 

18.  x^-6x^  -\-8x'^  -17x  +  10=0, 

19.  2;i;5-7;v*-9;t3  +  33.;i;2  +  17;i;~30=0, 


CHAPTER  XXXVI. 

DECOMPOSITION   OF   RATION AI.   FRACTIONS. 

835.  A  fraction  in  which  both  numerator  and  denom- 
inator are  rational  integral  functions  of  x  is  called  a 
Rational  Fraction. 

If  the  numerator  of  the  fraction  is  of  a  lower  degree 
than  the  denominator,  the  fraction  is  called  a  Proper 
Rational  Fraction. 

If  the  degree  of  the  numerator  is  equal  to  or  greater 
than  that  of  the  denominator,  the  fraction  is  an  Improper 
Rational  Fraction. 

An  improper  fraction  may  always  be  reduced  to  a  mixed 
expression ;  that  is,  to  the  sum  of  an  integral  expression 
and  a  proper  fraction. 

836.  The  problem  of  this  chapter  is  to  decompose  any 
given  rational  fraction  into  a  set  of  simpler  fractions  called 
Partial  Fractions,  whose  sum  is  equal  to  the  given 
rational  fraction. 

If  we  wish  to  decompose  an  improper  fraction,  we  first 
reduce  it  to  a  mixed  expression,  and  then  decompose  the 
resulting  proper  fraction. 

Since  any  improper  fraction  may  be  treated  in  this  way, 
we  will  confine  our  attention  in  the  discussion  which  fol- 
lows to  the  case  of  proper  rational  fractions. 

837.  We  will  first  decompose  a  rational  fraction  into 
the  sum  of  two  other  fractions,  then  the  second  of  the 
two  resulting  fractions  into  the  sum  of  two  more  fractions, 
and  so  on. 


670  UNIVERSITY   ALGEBRA. 

Let  4^^  denote  a  proper  rational  fraction  and  suppose 
a  root  a  occurs  71  times  and  no  more  in  Fipc).  Then  we  have 

Ax)^      Ax) 

F{x)      {pc-aYF^ix) 
where,  of  course,  F^{pc)  denotes  what  remains  of  F{x) 
after  removing  the  factor  {x—cCy,  and  evidently  F-^ipc) 
does  not  contain  the  factor  {x — a). 
Now,  whatever  A  stands  for,  we  have 

fix-)         ^AF,{x-)+f{x)-AF^ix) 
{x — aJF^  {pc)  (x — a)  "F^  (x) 

AF,(x)      _^/(x)-AF,(x) 


(x—a)  'F-^  ( Jf)       {x — a)  'F^  (;t:) 
A        ^  /(x)-AF,ix-) 


(x—ay  (x—ayF^(_x) 
As  the  sum  of  the  last  two  fractions  equals  the  given 
fraction  whatever  A  stands  for,  we  may  take  A  anything 
we  please.  Let  us  then,  if  possible,  select  A  so- that  the 
numerator  of  the  last  fraction  shall  contain  the  factor 
(x—a). 

Now,  this  numerator /(;«;)— ^/^i(;r)  is  some  function 
of  X,  and  hence,  if  it  contains  the  factor  x—a,  a  is  a  root 
of  Ax)—AF^(x)  by  Art.  471.  But  if  a  is  a  root  of 
/(x)—AF^(x)  this  expression  must  vanish  when  a  is 
substituted  for  x  by  Art.  736.  Hence,  we  must  have 
/(a)-AF,(a)=0. 

Hence,  ^  =  -Wt\ 

F^{a) 

an  expression  which  is  evidently  independent  of  x. 

With  this  value  of  A,   the  expression  /(;»:) -~^i^j,(.:r) 

contains  the  factor  (x—a),  and  hence  the  fraction 

f{x)-AF,{x) 

(x-ayF.ix-) 

can  be  reduced  to  lower  terms. 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.       67 1 

Now  representing  hy  f^(^x)  the  quotient  obtained  by- 
dividing /(.r)—^/^i(ji;)  by  {x—d),  we  have 


where  the  numerator  y^  is  independent  of  x  and  the  second 
fraction  of  the  right-hand  member  is  a  proper  rational 
fraction. 

In  exactly  the  same  manner  as  above,    the   fraction 

,     /^  ,  -r  mav  be  expressed  as   the  sum  of  two 

fractions.     Hence,  we  may  write 


(;^;-a:  ^^^i(^)      {x-ay-^      {x-ay-'' F^{pc) 
Again,  in  the  same  way,  we  have 


(3) 


Evidently  we  can  continue  to  obtain  successive 
equations  as  long  as  there  remains  any  power  of  {x~d) 
in  the  denominator  of  the  fraction  we  last  obtain. 
Hence,  making  successive  substitutions  in  equations  (1), 
(2),  (3),  etc.,  we  obtain 

{x-ayF^{x)      {x-ay^  {x-a)"-^'^'"  '^  x-a'^  F^{x) 

where  the  numerators  in  the  first  n  fractions  in  the  second 
number  are  all  independent  of  x,  and  the  last  fraction  is 
a  proper  rational  fraction. 

If  now,  a  root  b  occurs  s  times  and  no  more  in  F^  (x), 
then 

<f>{x)  ^         <j>(x) 

F,ix)      Cx^dyF,(xy 

in  which  F2  (x)  represents  the  quotient  obtained  by  divid- 
ing F,(x)  by  (x—dy. 


672  UNIVERSITY   ALGEBRA. 

In  exactly  the  same  way  as  above  it  may  be  shown  that 
<^(^)         _      B  B^  B^,       d>,(x} 

Evidently  the  process  pursued  thus  far  may  be  con- 
tinued until  all  the  factors  of  the  denominator  are  ex- 
hausted, when  the  decomposition  comes  to  an  end. 

/(x') 
Therefore,  any  proper  rational  fraction  ~~-(  can  be 

decomposed  into  a  set  of  partial  fractions  whose  numera- 
tors are  all  independent  of  x  atid  whose  denominators  are 
the  successive  powers  of  the  binomials  x — a,  x — b,  etc., 
(a,  b,  etc.,  being  roots  of  the  given  denominator);  the 
highest  power  to  which  any  binomial  appears  in  any 
denominator  being  the  power  to  which  that  same  bi- 
nomial appears  in  the  denominator  of  the  given  fraction. 

838.  We  will  next  show  that  there  cannot  be  two 
different  sets  of  simple  fractions  whose  sum  equals  the 
given  rational  fraction. 

Suppose,  if  possible,  that  the  rational  fraction  v^y-t  is 

F(x} 

equal  to  the  sum  of  each  of  the  two  sets  of  partial  fractions 

A  A,  A,^_,       6(x-) 

\  "T  7"      _N»4-i  "T  ••  •  •  -r— — -  -r 


(x—aY      {x—ay^^  x-r-a      F^{x) 

and     7 ^+7 TTZT  "^"  ■  *  •  ^ ^"cv~~7^>' 

{x—ay     {x—ay  ^  x—a     F ^{x) 

Each  of  these  sums  being  equal  to  the  given  fraction,  we 
have 

^  ^1  ,  A,,_^       <t>(x) 


(x—ay     (x—ay-'^  x—a     F^{x) 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.       673 

If  n  and  r  are  not  equal  to  each  other,  suppose  ri>r\ 
then  if  we  transpose  all  the  terms  of  the  first  member 

except  the  term  _^  and  represent  the  resulting  sec- 
ond member  by  ^^_^|f.^,^^^y  we  have 

(x^ay     (x—ay-^'ifCxy 

in  which  "^(x)  does  not  contain  the  factor  (x—a),  and  A 
is  independent  of  x.  If  we  put  x=  a  in  the  last  equation, 
we  get  ^  =  0.  Hence,  if  w>r,  we  must  have  A  —  ^. 
Hence,  if  ^^0,  n  cannot  be  greater  than  r. 

In  the  same  way  it  may  be  shown  that  if  n<ir,  we  must 
have  ^'=0.     Hence,  if  y4'=?^0,  n  cannot  be  less  than  r. 

Therefore,  as  n  cannot  be  either  greater  or  less  than  r, 
it  must  be  that  n=^r. 

Now,  writing  n  for  r  in  the  second  series  of  partial 
fractions,  equation  (1)  becomes 


{x—ay     {x—ay   1  x—a     F^  {x') 

A- A'  il/.ix-) 


{x—ay^{x-ay-^^       '^  x-a^F^Xx)         ^^ 


Hence, 


ix—ay     (x—ay-^^^(x) 


Hence,  A-A'=ix-a)^        , 

Now,  making  x=a,  it  is  evident  that  A—A^=0  or 
A=A\  Therefore  in  the  two  series  of  partial  fractions, 
the  two  terms  containing  the  highest  powers  of  (;r— a)  in 
the  denominators  are  equal  each  to  each. 

Now,  if  these  equal  terms  be  suppressed  from  equation 

43—  U.  A. 


6/4  UNIVERSITY   ALGEBRA. 

(1),  the  argument  repeated  would  show  that  the  second 
terms  of  the  two  series  of  partial  fractions  are  equal  each 
to  each,  and  so  on  to  the  end  of  each  series.  Hence,  the 
terms  of  the  first  set  of  fractions  are  equal  to  the  terms  of 
the  second  set  of  fractions  each  to  each.  Therefore,  the 
two  sets  of  partial  fractions  are  identical,  or  there  is  only- 
one  set  of  simple  fractions  whose  sum  equals  the  given 
rational  fraction.  This  fact  is  usually  expressed  by  say- 
ing that  a  rational  fraction  can  be  decomposed  in  only 
one  way. 

This  language,  however,  does  not  mean  that  there  is 
only  one  method  to  pursue  nor  even  that  the  fraction 
cannot  be  decomposed  into  partial  fractions  of  less  simple 
form,  but  that  in  terms  of  the  simple  fractions  described 
in  Art.  837  there  is  only  one  result  to  reach. 

839.  The  investigation  in  Art.  837  holds  whether  the 
roots  of  f{x)  are  real  or  imaginary,  but  if  some  of  the 
roots  are  imaginary  of  course  imaginary  expressions  will 
enter  some  of  the  partial  fractions.  We  may,  however, 
avoid  imaginaries  by  using  partial  fractions  of  a  less  sim- 
ple form.  In  the  investigation  of  Art.  837,  suppose 
a=a+tP,  then  we  know  that  there  is  another  root  of 
F(^x)  which  is  the  conjugate  of  a  +  z/S.  Suppose  d  is  this 
conjugate  root,  then  d^a—z/S.  We  know  that  these  two 
roots  occur  the  same  number  of  times  in  jFIx");  i.  e.,  r^n. 
Now,  because  a  rational  fraction  can  be  decomposed  in 
only  one  way,  the  values  of  the  numerators  of  the  various 
partial  fractions  will  not  depend  upon  the  method  used 
to  find  them  nor  the  order  in  which  they  are  found. 

In  Art.  837  we  found  A=  {}/'>  i.  ^.,  under  the  pres- 


ent  supposition  A=^~r-, — -r^- 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.      6/5 

Evidently,  however,  we  might  have  begun  by  finding 
the  numerators  corresponding  to  the  various  powers  of 
x—b.  To  find  B  we  would  proceed  exactly  as  above 
when  A  was  found  except  that  everywhere  the  sign  of  i 
would  be  changed.  Therefore,  the  value  of  B  is  the  same 
as  A  except  in  the  sign  of  i. 

Hence,  if  ^  =  #F^V 

If  we  represent  the  value  of  A  by  l+im,  the  value  of 
B  will  be  / — t7n. 

Therefore  two  of  the  partial  fractions  in  the  above  series 
become  in  the  case  here  considered 

l+im  ^         l—im 

and 


{x—a—i^y  (x—a+ipy 

Now  it  is  readily  seen  that  the  sum  of  these  two  frac- 
tions is  real. 

For,  let  Cx—a^tpy^L-ht'M, 

then  (ix—a-\-zfiy=L—zM, 

l+nn        l—im  __2{/L  +  7nM) 
L-ViM^ L^M"    Z,2+^"2— 

Now,  since  {x—a—ipy'=L  +  iM,  it  is  evident  that  L 
is  a  function  of  x  of  the  nih  degree,  and  M  a  function  of 
X  of  the  {n—l^st  degree.  Therefore,  the  sum  of  the  two 
partial  fractions  is  a  fraction  whose  numerator  is  of  the 
degree  n  and  whose  denominator  is  of  the  degree  27t. 
The  denominator  is  in  fact  {x'^—2ax-\-a'^-\rP'^yy  hence  we 
may  write 

A  ,  B  <t>^"{x) 


where  <l>'\(x)  is  of  the  degree  n. 


(1) 


t^6  UNIVERSITY    ALGEBRA. 

In  precisely  the  same  way  the  partial  fractions 
^     -  and  ^ 


when  added  together  give  a  real  fraction  whose  numera- 
tor is  of  the  degree  n—\,  and  whose  denominator  is  of 
the  degree  2(7Z  — 1).  The  denominator  in  this  case  is 
(;t;2—2a;ir4-a2+/3 2 )'*-!,  and  hence  we  may  write 

-^1 + ^ = ^^Z(fL_  .2) 

In  a  similar  way  we  obtain 

^.  .  Bj_ <^{^ ..^ 

(;t:_a-/iS)''-2"^(;t--a4-2W"-2  (;r2-2a;f -f  a^  4-/52)«-2  W 
Evidently  this  process  of  grouping  partial  fractions 
together  as  has  just  been  done  may  be  continued  until  all 
the  corresponding  partial  fractions  are  grouped  together 
into  real  sums.  When  this  is  done,  instead  of  the  origi- 
nal partial  fractions  corresponding  to  the  roots  a-fz^  and 
o.—i^  each  occurring  n  times  in  the  original  denominator, 
we  have  the  partial  fractions 

If  now  these  fractions  be  reduced  to  a  common  denom- 
inator and  added  we  obtain  a  single  fraction  whose  de- 
nominator is  (.r2  — 2a.r  +  a2-f-/?-)"  and  whose  numerator 
is  a  real  function  of  x  of  the  wth  degree. 

xCf) 

(;»;2-2a;r  +  a2+y82)n 

Divide  the  numerator  by  (;r2  — 2a;f+a2-f  ^2^^  ^^^  1^^ 

Xi(^)  denote  the  quotient  and  Cx-\-D  the  remainder,  then 
X(;tr)  =  (;r2-2a;t:  +  a2+/32)^^(;^)4_C-_f.^. 


Hence, 


(;»;2~2a;t:  +  a2+/32)« 

Cx^-D ^ XiW 


(;«;2_2cu;-|-a2+/32)«    '   (;^2_2a;t:  +  a2+^2)«- 


'-1 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.      ^TJ 

The  same  process  may  be  repeated  upon  the  second 
fraction  in  the  right  member  and  repeated  again  in  the 
second  fraction  of  the  resulting  right  member,  and  so 
on  until  a  fraction  is  obtained  whose  denominator  is 
{x-—^a.x-\-aP'-\-^'^-''),  We  thus  obtain  ;z  partial  fractions 
whose  numerators  are  of  the  first  degree  and  whose  denom- 
inators are  the  successive  powers  of  {x'^—^o.x+aP'-\-P'^^. 

Hence ^— 

^  Cx+D  C,x-\-D^ 

C^X  +  D^ Cn-xX-\-Dn-^ 

'     /  ^.9.         O r    -.2     I     0'2\H—'i  "t"  •    •    •   "t" 


Reviewing  now  article  837  with  the  present  article,  we 
conclude  that  to  each  real  root  occurring  n  times  in  the 
denominator  of  a  given  proper  rational  fraction  there  cor- 
responds n  partial  fractions  of  the  form  given  in  the  first  n 
fractions  of  the  second  member  of  the  equation  in  Art. 
837,  and  to  each  pair  of  imaginary  roots  occurring  r  times 
in  the  denominator  of  the  given  fraction  there  corresponds 
r  partial  fractions  of  the  form  given  in  this  article^ 

This  may  be  otherwise  expressed  by  saying  that  to  a 
factor  (:r— a),  which  occurs  n  times  and  no  more  in  the 
given  denominator,  there  corresponds  n  partial  fractions 
of  -the  form  given  in  Art.  837,  and  to  an  irreducible 
quadratic  factor  x'^  —Icxx-^oP-  -\- ^'^ ,  which  occurs  r  times 
in  the  given  denominator,  there  corresponds  r  partial 
fractions  of  the  form  given  in  this  article. 

DETERMINATION  OF  NUMERATORS. 

840.  Having  found  the  form  of  the  partial  fractions 
into  which  a  given  fraction  can  be  decomposed,  it  remains 
now  to  determine  the  values  of  the  various  numerators. 


6/8  UNIVERSITY   ALGEBRA. 

We  will  first  take  a  particular  example.  After  explain- 
ing this,  the  statement  of  the  method  to  be  pursued  in  any 
case  will  be  readily  understood.     Let  us  decompose  the 

Assume 

(^x+i)\x'-\-x^iy   (^x-^iy^ x^i    {^xf^x+iy   x'-\-x+i 

Reducing  the  partial  fractions  to  a  common  denomin- 
ator, adding,  and  clearing  of  fractions,  we  obtain 
x'^-'Zx'-'2=-A(^x'^  +x+\y  +A  ^{x+l)(x'^  +x+iy 
+  {Bx-]-C)(,x+\y 

+  (^B,x+C,')(x+l)\x''+x+r). 
Arranging  the  second  member  of  this  equation  accord- 
ing to  powers  of  x,  we  have 

x^'-^x—2={A^+B^)x'^-\-(^A+ZA^+ZB^  +  C^')x^ 
+  (2A+bA^+B-^AB,  +  '^C^)x^ 

+  (2A+ZA^+B+2C^B^+ZC,)x 

Now  as  the  sum  of  the  assumed  partial  fractions  equals 
the  given  fraction  whatever  value  be  given  to  x,  the 
equation  last  written  is  an  identical  equation,  and  hence, 
by  Art.  596,  the  coefficients  of  like  powers  of  x  on  the 
two  sides  of  the  sign  of  equality  are  equal  each  to  each. 
Hence,  by  equating  coefficients,  we  have 

A,+B,=0. 

^  +  3^,+3^,  +  Ci=0. 

2^+5^1 4- i5+4i5i+3Ci==0. 

3^+5^1 +2Z?+C+3i9i+4Ci  =  l. 

2^  +  3^1 +i5+2C+i5i4-3Ci  =  -3. 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.       679 

Solving  these  equations,  we  obtain  the  values 
^=3.         A,==     1. 
i?=2.         i?,  =  -l. 
C=-2.      Cx  =  -3. 

Introducing-  these  values  in  the  assumed  set  of  partial 
fractions,  we  have 

(x+iyix^+x-hiy     C;^+l)2"^;c+l 

2;c-2  -'X-S 

'^(x^-hx+iy'^x'^+x+l 
The  method  pursued  in  this  example  will  render  the 
following  directions  readily  understood. 

First.  If  the  fraction  is  improper,  reduce  to  a  mixed 
expression  and  consider  then  the  resulting  proper  rational 
fractions. 

Second.  If  this  fraction  is  not  in  its  lowest  terms,  re- 
move all  fractions  common  to  numerator  and  denomin- 
ator. There  will  then  remain  a  proper  rational  fraction 
in  its  lowest  terms. 

TJiird.  Equate  this  proper  rational  fraction  to  the  sum 
of  a  set  of  partial  fractions  of  the  forms  described  above 
with  undetermined  numerators. 

Fourth.  Clear  the  resulting  equation  of  fractions  by 
multiplying  by  the  denominator  of  the  fraction  in  the  first 
member. 

Fifth.  Equate  coefficients  of  like  powers  of  x  in  the  two 
members  of  the  resulting  equations. 

Sixth.  Solve  the  resulting  equations  and  thus  find  the 
numerators  of  the  assumed  set  of  partial  fractions. 

841.  The  above  method  may  be  somewhat  shortened, 
especially  when  the  denominator  of  the  given  fraction  has 


680  UNIVERSITY   ALGEBRA. 

no  imaginary  roots;  that  is,  when  it  has  no  irreducible 
quadratic  factors.     This  is  done  as  follows : 

Multiplying  by  /^(:r),  we  obtain 

fCx)  =  Aoct>(x-)+A,(x-a')<f>(ix)+  .  .  .  +(;c-a)'',/.(;r). 

Since  this  equation  is  true  for  all  values  of  x,  let  x=a. 
Then  all  the  terms  in  the  second  member  except  the 
first  term  vanish,  and  we  obtain 

Hence  ^0  =  4-^ 

</)(a) 

If  now  the  term  ~t4^M  be  transposed,  we  obtain 

+  •  •  -+(x-^ayxl;(x). 

Both  members  of  this  equation  are  evidently  divisible 
by  x—a^  and  when  thus  divided,  x  may  again  be  placed 
equal  to  a  and  A^  determined. 

By  repeating  this  process  we  can  determine  the  numer- 
ators of  all  those  partial  fractions  whose  denominators  are 
powers  of  x—a.  In  a  similar  way  the  numerators  of 
the  other  partial  fractions  can  be  found, 

EXAMPLES. 

Decompose  the  following  fractions  : 

x'^  —  Qx  +  l  2x^  +  5x'^+4x+S 


x^-6x'^  +  llx-Q  ^'     (;»;+l)2(jt;2_^l) 

x'^+6x-h7  Sx'^+x-j-Q 

^'  x^-^6x''-i-nx+Q  •   (:r-l)2(.r2-l)* 

x—1    Sx^+x'^-^x+l 

5x^-5x-h2  o    2x^  +  2x^-  +  2x-h2 

o. 


{,x+lXx^  +  l)  •  x*  +  l 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.      68 1 
RAPID   METHOD. 

842.  The  methods  already  described  for  decomposing 
rational  fractions  are  the  ones  usually  given  in  text  books, 
but  if  the  denominator  of  the  given  fraction  contains  some 
factor  to  a  high  power  the  work  of  finding  the  numera- 
tors is  very  tedious,  and  for  the  decomposition  of  such 
fractions  some  less  tedious  process  is  much  to  be  desired. 
Applied  to  some  fractions  the  process  which  follows  is 
several  times  more  rapid  than  those  already  explained. 

lyCt  the  fraction  to  be  decomposed  be  represented  by 

Transform  both  numerator  and  denominator,  as  in  Art. 
762,  by  substituting  y-{-a  for  Xy  and  let  the  transformed 
fraction  be  represented  by 

yXb,y'-^b,y"-^+b^y"-^'{'  •  •  •  +b„) 
Assume  this  fraction  equal  to 

Clearing  of  fractions,  we  have 

«oy+^iy^  +  ^2y~^+  •  •  •  +^r 

^A,{b,y^+b,y^-^^""^b,:)+A^y{b,y-+b^y--^  +  -+b,:) 
-\-A,y\boy''  +  b,y'^-^+.  .  •  +b„)+  •  .  •  +yF(y). 
Putting  y=Oy  we  obtain 

^o^„=^^or^o=-^' 
that  is,  Aq  equals  /ke  absolute  term  of  the  first  member  di- 
vided by  the  absolute  term  of  the  second  member. 

If  now  the  term  A^{b^y''-\-b^y''-^+  .  .  .  +3„)  be  trans- 
posed and  the  resulting  equation  divided  by  y,  the  new 
absolute  term  of  the  first  member  will  be  ar-Y—A^b,^_^, 
and  the  new  absolute  term  of  the  second  member  will 
be^i^«. 


682  UNIVERSITY   ALGEBRA. 


Hence,  -^i=" 


b.. 


In  a  similar  manner  we  obtain 

^, ^—.  -. 

And  in  general 

Thus  it  is  seen  that  after  the  fraction  has  been  trans- 
formed to  a  scale  of  y  the  calculation  of  the  numerators 
to  the  various  powers  of  y  in  the  denominator  may  be 
very  readily  obtained,  each  numerator  being  expressed 
in  terms  of  all  the  previous  numerators.  The  transfor- 
mation of  the  fraction  to  the  scale  o\  y  is  best  accom- 
plished by  synthetic  division,  the  work  being  arranged 
as  in  Horner's  method. 

We  will  illustrate  this  process  by  decomposing  the 
fraction 

If  we  transform  the  numerator  as  in  Art.  762  by  letting 
x=y+Sy  the  work  will  be  arranged  as  follows: 


-8 

+  21 

-13 

-5 

+  6  1 

5 

—2 

1        ^ 

-^— 

1      1 

Hence  the  numerator  of  the  new  fraction  is  y^+y^+5, 

;^;3_gy2^2U'•— 13_  j/^+jK^+5 

Hence,  (^^_sy\x-2y    ""j/i^^+l)'* 

Expanding  (^+1)^  this  last  fraction  becomes 

j/g+j^/^  +  5 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.       683 
Assume  this  fraction  equal  to 


+41+ 


Then,  by  the  formula  already  explained,  we  obtain  the 
following  values : 
Aq^j=5.     Since  d„  is  in  this  case  1,  we  need  not  write 

the  denominator,  and  in  the  following  equations 

the  denominator  is  omitted. 

^2  =  1-5x28  +  40x8  =  181. 
^3  =  1  -  5x56  +  40x28  -  181x8  =-607. 
A^=0  -  5x70  +  40x56  -  181x28  +  607x8  =  1678. 
^.=0-5x56  +  40x70-181x56  +  607x28 
-1678x8  =-4044. 

^g=0-5x28  +  40x56-  181x70  +  607x56 

-  1678  X  28  +  4044  x  8  =8790. 
A,=0  -5x8  +  40x28-181x56  +  607x70 

-  1678  X  56  +  4044  x  28  -  8790  x  8  =  -17642. 
^3=0  —  5x1  +  40x8-181x28  +  607x56 

-  1678  X  70  +  4044  x  56  -  8790  x  28 

+  17642x8=32250. 

A^=0  -  5x0  +  40x1  -  181  x8  +  607x28 

-  1678  x5(j  +  4044  x  70  -  8790  x  56 

+  17642  X  28  -  32259  x  8  =  -51636. 

If  now  we  restore  the  value  of  jj/,  viz.:  x—3,  we  have 
all  the  partial  fractions  whose  denominators  are  powers 
of  x—S. 

To  obtain  the  numerators  of  those  partial  fractions 
whose  denominators  are  powers  of  x—2,  we  begin  with 
the  given  fraction  as  though  none  of  the  partial  fractions 


684  UNIVERSITY   ALGEBRA. 

were  determined.  We  transform  the  numerator  of  the 
original  fraction  to  a  scale  of  z  where  ^=^+2,  arrang- 
ing the  work  as  follows : 


-8 

+21 

-13 

-6 

9 

6 

-4 

1 

-2 

Hence  the  numerator  of  the  new  fraction  is^*^— 2^2+5'+5. 
Hen  OP = ■ . 

^        {^x-6y\x-iy        (^-1)10^8 

Expanding  (^—1)^^,  this  last  fraction  becomes 

Z^  —2Z^  +  2^  +  5 

^8,^;gio— 10^9 -I- 45^8-12027+21026-25225+21024-12023+4502-100+1) 
Assuming  this  fraction  equal  to 

B,     B,     B,     B,     B.B,     B,     B,       <f>{z) 
z^'^  z'^       z^      z'^'^  z^^  z^'^  z''        z  "^(^-1)10* 
Then,  by  the  formula  already  explained,  we  readily 
obtain   the    following   values,   where   the   denominator 
being  1  is  in  each  case  omitted. 

^^  =  1  +  5x10=51. 

^2  =  -2 -5x45 +  51x10=283. 

.     ^3  =  1  +  5  X 120  -  51 X  45  +  283  x  10  =1136. 

i?4=0  -  5x210  +  51x120  -  283x45  -  1136x10 
=  3695. 

i?g=0  +  5  X  252  -  51 X  210  +•  283  x  120  -  1136  x  45 

+  3695x10=10340. 
^g=0  -  5  X  210  +  51 X  252  -  283  x  210  +  1136  x  120 

-  3695  X  45  +  10340 x  10  =25817. 
^^ =0  +  5  X 120  -  51 X  210  +  283  x  252  — 1136  x  210 

+  3695  X 120  -.  10340  x  45  +  25817  X  10 

=  58916. 


DECOMPOSITION  OF  RATIONAL  FRACTIONS.     685 

If  now  we  restore  the  value  of  z,  viz.:  x—^,  we  have 
all  the  partial  fractions  whose  denominators  are  powers 
oi  X — 2,  and  since  the  partial  fractions  whose  denomina- 
tors are  powers  of  x — 3  have  already  been  found,  we  have 
completely  decomposed  the  given  fraction.  Hence  we  have 
the  following: 


607  1678  4044  8790         17642^ 

{x-zy^{x-zy    {x-zy'^ {x-zy   {x-zy 

82259      51631  .        5        .       51       .      283 


{x—zy    x-z  '  (;r-2)8  '  {x-iy  '  {x-iy 

1136  8695         10340        25817      58916 

■^(;r-2)5'^(;tr-2)4"^(;r~2)«"^(;i;-2)2+^~2* 
We  have  purposely  chosen  an  example  from  which, 
by  ordinary  methods,  the  most  indefatigable  calculator 
would  shrink. 

Other  rapid  methods  of  decomposing  rational  fractions 
are  given  in  the  Cambridge  and  Dublin  Mathematical 
Journal,  vol.  Ill,  and  in  the  Mathematician,  vol.  III. 


CHAPTER  XXXVII. 

GRAPHIC  RKPRKSKNTATION  OF  EQUATIONS. 

843.  The  graphs  of  simple  equations  in  two  variables, 
of  systems  of  two  such  equations,  and  of  equations  of  the 
form  y=ax^  +  dx-\-Cy  have  already  been  considered:  see 
pages  259-2G6.  It  now  remains  to  consider  the  graphs  of 
equations  of  higher  degree  and  some  of  the  general  prop- 
erties of  such  graphs. 

EQUATIONS  OF  THE  FORM^=/(;r). 

844.  Every  equation  of  the  form  y=f{x)  is  the  equa- 
tion of  a  curved  line.  For  no  matter  what  value  be  as- 
signed to  X,  y  will  have  a  single  value,  and  the  assigned 
value  of  X  and  this  resulting  value  of  y  will  together 
locate  a  point.  But  as  ^  is  a  variable,  the  point  {x,  y) 
is  not  fixed  in  position,  hntis  a  moving- poinL  Also,  since 
/(x)  is  continuous  by  Art.  748,  if  we  make  x  change 
continuously,  then  y  will  change  continuously,  and  the 
point  must  describe  a  contmuous  path  or  airve, 

845.  As  a  particular  instance,  consider  the  equation 

jK=^^  — .6;t:2  — .88:r+.192. 
A  table  of  values  may  be  formed  by  assigning  values 
to  x  and  computing  the  corresponding 
values  of  y,  as  in  Art.  387.  The  roots 
of/(;t:)  have  been  inserted  in  their  proper 
places  in  the  table.  The  graph  is  shown 
in  fig.  21.  The  distance  from  the  origin 
to  A,  B,  and  C,  respectively,  are  the 
three  roots  oif{pc).  At  E  is  represented 
a  minimum  value  oi  fix)^  and  at  E'  is 
represented  a  maximum  value. 


X 

y 

—   1 

-  .528 

-  .8 

0 

0 

+  .192 

+  .2 

0 

+    1 

-  .288 

+  1.2 

0 

+    2 

+4.032 

GRAPHIC  REPRESENTATION  OF  EQUATIONS.  68/ 


'X'  A/ 


.\^ 


Ic  X 


Ir 

Fig.  21. 

846.  In  drawing  the  graphs  of  equations  it  will  be 
found  frequently  that  the  values  ofy  are  very  large  com- 
pared to  the  corresponding  values  of  ;r,  thus  the  difficulty 
arises  of  conveniently  representing  the  graph  on  a  piece 
of  paper  of  the  usual  size.  This  difficulty  can  be  avoided, 
however,  by  using  a  smaller  tmii  of  measure  for  the  y's 
than  for  the  x's ;  for  example :  an  inch  as  the  unit  of 
measure  for  x  and  one-sixteenth  of  an  inch  as  the  unit  of 
measure  for  y.  The  proper  adjustment  in  each  instance 
must,  of  course,  be  determined  by  circumstances.  Al- 
though the  graph  is  condensed  in  one  dimension  by  this 
process,  yet  many  of  the  essential  properties  of  the  graph 
remain  unchanged. 

847.  By  the  Center  of  a  curve  is  meant  a  point  such 
that  all  lines  passing  through  that  point  and  terminated 
by  the  curve  are  bisected  by  the  point. 

Consequently  if  a  curve  has  but  one  maximum  and  but 
one  minimum  point,  the  center,  if  there  be  one,  must  lie 
half  way  between  these  points. 

A  curve  is  symmetrical  with  respect  to  its  center,  and 
if  a  curve  be  symmetrical  with  respect  to  a  point,  that 
point  is  the  center  of  the  curve. 


688  UNIVERSITY    ALGEBRA. 


KXAMPIyKS  AND  PROBI^KMS. 


Draw  the  graph  of  each  of  the  following  equations. 
In  some  cases,  as  in  4  and  5,  fractional  values  must  be 
substituted  for  x  in  order  to  determine  the  form  of  the 
graph. 

1.  y=x^—Sx'^—6x+8.         5.  j^=x^  +  Sx'^  +  2x, 

2.  y=x^—5x'^+4:.  6.  y==x^  +  2x+S. 

3.  jy=x^—5x'^—4x+20,         7.  jy=x^—Sx. 
/^,  j;=x^—Qx^  +  lix—6.         S,  jy=x^+Sx. 

g.  y=2x^—7x^+4:x+4t. 

10.  In  the  graph  of  y=^f{pc),  what  represents  the  real 
roots  of /(;»:) =0? 

11.  Changing  the  absolute  term  oif{x)  has  what  eiOFect 
on  the  graph  of  y—f\xy> 

12.  If /(:i:)  is  a  cubic,  and  a^,  ag,  and  a^  represent  its 
three  roots,  show  that  the  center  of  y==f(x')  is  at 
|(a^  -f  0-2  +^3)  units  from  the  j^-axis  and  that  the  numerical 
distance  from  maximum  or  minimum  point  to  vertical  line 

through  center  is  (ai^+a2^+^3^"~^i<*2~^i^3"~^2*3)^* 

13.  Determine  the  condition  thsity=x^+ax^  +  dx+6 
has  710  maximum  or  minimum  points. 

14.  If  ^^  =  8<^,  discuss  the  maximum  and  minimum 
points  of  y=x^+ax'^  +  dx+Cy  and  the  form  of  the  curve. 

15.  Show  the  effect  of  Horner's  transformation  on  the 
graph  of  jK=/W. 

In  Horner's  transformation,  x-h^  is  substituted  for  x  in  /{x). 
That  is,  all  the  x's  in  /{x)  are  increased  if  A  is  negative,  and  all  tha 
x's  in  /(x)  are  decreased  if  /i  is  positive.  In  the  first  case  the  graph 
is  moved  bodily  to  the  right  and  in  the  second  case  it  is  moved  to  the 
left. 


GRAPHIC  REPRESENTATION  OF  EQUATIONS.     689 

16.  Show  that  jr=x^-^kx  represents  all  /or?ns  of 
curves  represented  by  y=x^ +ax^  +  dx+c. 

By  Horner's  method,  Art.  7C2.  the  second  term  of  x^-{-ax^ -\-dx+c 
may  be  removed.  By  example  15  this  moves  the  curve  to  the  right 
or  left  without  changing  its  form.  Then  by  dropping  the  absolute 
term,  by  example  11,  the  curve  is  raised  or  lowered,  depending  upon 
the  sign  of  the  absolute  term.     We  then  have  an  equation  of  the  form 

17.  Show  that  the  graph  of  any  cubic  equation  of  the 
form  y=/(x)  is  symetrical  with  respect  to  one  of  its 
points,  which  is  the  point  we  have  termed  the  center. 

Move  the  curve  until  its  equation  is  of  the  form  y=x^-{-/:x. 
Suppose  x' ,y'  is  a  point  on  the  curve.     Then  we  have 

y'=x'^  +  /:x'.  (1) 

Putting  —x'  for  x  and  —y'  for  y  in  y=x^-{-kx,  we  get 

—y'  =  —x'^—kx',  (2) 

which  is  the  same  as  (1).  Therefore,  ilx' ,  y'  is  a  point  on  the  graph, 
the  point  —x',  —y'  is  also  a  point.  Therefore,  the  curve  is  symmet- 
rical with  respect  to  the  origin.  Therefore,  by  Art.  847,  the  origin  is 
the  center  of  the  curve. 

18.  Find  the  position  with  reference  to  the  j/-axis,  of 
the  center  and  of  the  maximum  and  minimum  points  of 
y=:x^+ax'^-{-bx+c. 

Here/(-r)=x3  +  ^x8  +  ^-^+<r, 
then  /'(x)=3x«-|-2^^+^. 

By  Art.  7G4,  the  roots  of  /'  {x)  determine  the  maximum  and  mini- 
mum points  of  the  graph. 

Solving  3.Ar2  +  2^x  +  /^=0, 

we  get  x—l{-a±\/a^-.^d) 

These  values  are  the  distances  of  the  points  in  question  from  the 
^'-axis.     That  x=l{-a-^\/<i''^-'^^) 

is  the  minimum,  and  that 

x—\{—a  —  \/a'-  —  '6if) 
is  the  maximum,  may  be  determined  by  substituting  in  /"{x),  as  in 
Art.  7G5.     It  is  plain  thnt  the  maximum  and  minimum   points  are 
each  distant,   numerically,  i\/tz=«-ii^  units  from  a  line   —^a  units 

44—  U.  A. 


690  UNIVERSITY   ALGEBRA. 

from  the  jj/-axis.  Therefore,  the  center  of  the  curve  is  — J<7  units 
from  the  ^-axis,  as  the  center  is,  by  Art.  847,  half  way  between  the 
maximum  and  minimum  points. 


In  fig.  21,  OD=i{-a-\.^/a^-^^),  OD=i{-a-\/a^-Bd), 
BDzsis/a^^^b,  BD'  =  —  y/a^—60,  and  03=— ^a. 

Draw  the  graphs  of  the  following  equations: 

19.  y=x^  +  2x^'-^x'^—4:X+2, 

20.  y=.x^—4:X^+x'^+1x—Z, 

21.  y^-x^'-x^  —  l^x^+x-]-!^. 

22.  y=x^  — "1x^  —  1  x'^Sx-{-lQ. 

23.  j/=2.;t:5-7.:r*-9ji;5+ 33.^2 +  17.^-30. 

24.  y=x^—^x^-lQx-\-AS. 

25.    y=x^ — bx.  26.     y=x^  +  5x. 

27.  What  is  the  difference  between  the  graphs  when 
y=f{pc)  is  of  odd  degree,  and  when  j^=/(:r)  is  of  even 
degree? 

28.  What  shows  that  every  f{pc)  of  odd  degree  has  at 
least  one  real  root? 

29.  How  many  times  can  a  line  parallel  to  the  j/-axis 
cross  the  graph  of  jK=/(^)? 

30.  How  many  times  (at  most)  may  any  straight  line 
intersect  the  graph  oi  y^^fixy, 

31.  How  are  two  equal  roots  of  /(.:r)  represented  in 
the  graph  of  y^fix)! 

32.  What  shows  that  imaginary  roots  enter  in  pairs? 

33.  What  is  the  greatest  number  of  maximum  and 
minimum  points  in  the  graph  of  jV=/(-^)? 


GRAPHIC  REPRESENTATION  OF  EQUATIONS.  69I 

34.  Form  the  derivative  of /(x)  in  any  of  the  exam- 
ples 1-9  and  19-26,  and  draw  the  graph  of  y=f'(x)  on 
the  same  axis  as  the  original  curve.  Note  the  geomet- 
rical meaning  of  the  theorem  of  Art.  766. 

35.  Draw  the  graphs  of  the  following  equations,  and 
note  the  change  in  the  graph  while  passing  from  three 
unequal  to  three  equal  roots : 

(a)  y=x^—\x;    (b)  y=x^—,01x;    (c)  j/^x^ —Ox. 

36.  Draw  the  graphs  of  the  following  equations,  and 
note  the  peculiarity  of  the  graph  when  there  are  several 
equal  roots : 

(a)j;=(^-l)3;        (b)j^=(^-l)^       (C)^=(.^-1)^ 
GRAPHS  OF  EQUATIONS  OF  TH:^  FORM/(i»,l/)=0. 

848.  We  shall  now  give  our  attention  to  the  graphs 
of  a  few  common  equations  of  the  form  f{x^  J>')=0.  For 
example,  consider  the  equation 

The  following  table  of  values  may  be  easily  obtained 
by  assigning  values  to  x  and  computing 
the  corresponding  values  oiy.  For  each 
value  of  x  we  get  two  points  of  the  graph, 
one  below  and  one  above  the  .r-axis. 
The  points  are  seen  to  lie  upon  a  circle. 
That  all  points  of  the  graph  must  lie  upon 
this  circle  may  be  seen  from  the  equation 
itself,  for  since  x'^-\-y'^=-2h  and  since  x 
and  y  are  always  the  legs  of  a  right  tri- 
angle, the  hypothenuse  of  the  same  must 
always  be  5.  That,  is  all  points  of  the 
graph  must  be  5  units  from  the  origin, 
and  consequently  constitute  a  circle  with 
radius  equal  to  5. 


X 

y 

-6 

±V- 

•11 

-5 

0 

-4 

±8 

-3 

±4 

—2 

±4.6 

-1 

±4.9 

0 

±5 

+  1 

±4.9 

+  2 

±4.6 

+  3 

±4 

+4 

±3 

+  5 

0 

+6 

±1/- 

-11 

692  UNIVERSITY   ALGEBRA. 

KXAMPI^KS. 

Draw  the  graphs  of  the  following  equations  : 


I. 

x^+y-^iQ. 

5.     ;t;2+4y2  =  36. 

2. 

x-'—j^=25. 

6.     36j;2  +  100j/2  = 

3- 

xy=12. 

7.     ;ir2— 4^2=36.1 

4- 

y^=^9,x. 

8.     ^2  =  6;t:2+2j;* 

9- 

(^ 

—2)2 

+  (;*r-3)2=25. 

10. 

y' 

=6;f' 

'+ArS. 

II. 

x^ 

+J/3: 

=27. 

849.  Names  are  given  to  several  of  the  graphs  of  the 
above  set  of  equations.  Thus :  1  and  9  are  Circles  ; 
2  and  3  are  Rectangular  Hyperbolas ;  5  and  6  are 
Ellipses;  7  is  a  Hyperbola;  4  is  a  Parabloa. 

GRAPHS  OF  QUADRATIC  SYSTKMS. 

850.  The  theorems  of  Arts.  398-401  are  elucidated 
in  a  striking  manner  when  we  apply  the  graphic  method. 

Consider  the  system 

:r2+;/2=34  (1)) 
;ry=15  (2)f  • 
Drawing  the  graph  we  find  a  circle  and  rectangular  hyperbola  in- 
tersecting in  four  points.  The  co-ordinates  of  these  points  of  intersec- 
tion constitute  the  solution  of  system,  for  since  a  point  of  intersection 
is  found  on  both  graphs,  the  co-ordinates  of  such  point  must  satisfy 
both  equations. 

The  natural  step  in  the  solution  of  the  system  is  to  add  twice  (2)  to 
(1),  thence  deriving  the  system 

^2  4-2x;/+7^=64     (3)), 
2^/-=:i0     (4)f''- 
The  graph  of  this  system  is  the  hyperbola  above  and  two  parallel 
"northwesterly"  straight  lines,  intersecting  the  hyperbola  in  the  satne 
points  in  which  it  ivas  cut  by  the  ci  cle. 

Thus  the  transformation  of  the  equations  of  the  system  has  changed 
the  graph,  but  has  preserved  the  points  of  intersection  unchanged. 
It  is  the  location  of  these  points  of  intersection  that  we  seek. 


GRAPHIC  REPRESENTATION  OF  EQUATIONS.     693 

The  next  natural  step  in  the  solution  is  to  subtract  four  times  (4) 
from  (3),  thence  deriving  the  system 

x'^+2xy+y^=6i     (5)) 
x^-2xy4.y^=  4     (6)  f 
The  graph  of  this  gives  us  the  same   "northwesterly"  lines  as  be- 
fore, but  two  parallel  "northeasterly"  lines  in  place  of  the  hyperbola. 
The  points  of  intersection  of  these  lines  are  the  same  points  noted 
above. 

The  next  step  in  the  solution  is  to  write 
x+y=±S     {7)  I  J 
x-y=±2     (8)r- 
which  is  exactly  the  same  as  c  and  gives  the  same  graph.    From  this 
we  derive  the  four  system 

j^=3r^'    ^=5r*'    ;'=~3r3-    >^=-5r*- 

We  say  that  the  original  system  is  equivalent  (see  Art.  396)  to  these 
four  systems.  Each  of  the  systems  ^j,  ^g,  e^,  and  e^  consists  of  two 
straight  lines,  parallel  respectively  to  the  ^-axis  and  jr-axis.  The  four 
points  of  intereection 

(5,  3)     (3,  5)     (-5,  -3)     (-3,  -5) 
given  by  these  systems  constitute  the  four  solutions  of  the  original 
systems. 

Next  consider  the  linear  quadratic  system 
x»-^y^=25     (1) 
X  -{-y  =  1     (2) 
Squaring  (2), 

x«4-;''=25     (3) 
x*  +  2yx-\-y^=i9    (4) 

Subtracting  (3)  from  (4), 

x^+y*=2o  (5) 

2xy  =24  (G) 
Adding  and  subtracting  (5)  and  (6), 

x^-2xy-\-y^=  1  (7) 

x^-\-2xy-j-y^=49  (8) 

Taking  square  roots, 

x-y=±l     (9) 
x-\-y=:±7  (10) 
Correct  solutions:  Introduced  solutions: 

X=:zni     r  X=:^\     .  X=-^\  X=-\\      . 

^==4^1-  y^^\J^-  y=-\\J^'       ^=_3fA- 

If  we  draw  the  graphs  of  (1)  and  (2),  we  shall  have  a  straight  line 
and  circle  intersecting  at  two  points.  The  co-ordinates  of  these 
points  of  intersection  constitute  the  solution 


K 


d. 


694  UNIVERSITY    ALGEBRA. 

If  we  draw  the  graph  of  system  [d)  we  shall  find  a  circle  and  twa 
parallel  straight  lines  intersecting  it.  Of  the  four  points  of  intersec- 
tion, two  are  introduced  by  squaring  (2).  System  {c)  gives  a  rectang- 
ular hyperbola  and  circle  meeting  in  the  same  four  points  as  before. 
System  d  (and  e  which  is  identical  with  it)  give  two  sets  of  parallel 
straight  lines,  cutting  the  axes  at  an  angle  of  45°,  and  intersecting 
each  other  in  the  same  points  as  above,  The  systems  f-^.f^^f^^  3-^^ 
f^  give  us  lines  parallel  to  the  respective  axes,  intersecting  at  the 
same  four  points  noted  above. 

Since  the  solution  to  any  system  of  two  equations  con- 
sists of  one  or  more  systems  of  the  form 

x=^a  \ 

y=b] 

it  follows  that  no  matter  what  the  graph  of  the  original 
system  may  be,  the  solution  results  in  a  graph  consisting 
of  straight  lines  parallel  to  the  x  and  y  axes. 

KXAMPI.KS. 

In  like  manner  to  the  above,  the  student  may  draw  the 
graphs  of  the  following  systems  and  consider  the  solutions: 
I. 


2. 


xy=  4. 

3- 

x+j=6. 
xy—b. 

;ir2+_j/2=20. 

X  +j/=  6. 

4- 

X  -y  =   2. 

5- 

x-" 

xy 

=  5. 
=6. 

CHAPTER  XXXVIII. 

DETERMINANTS. 

851.  I^et  US  take  two  simultaneous  equations  of  the 
first  degree  and,  from  them,  find  the  values  of  ;i;  and  y. 

a^x+b^y^c^  (1) 

a^x-^b^y=C2  (2) 
Multiply  (1)  by  b^  and  (2)  by  b^,  whence 

a^b^x-^b^b^y^c^b^  (3) 

a^b^x-{'b^b^y=c^b^  (4) 
Subtracting  (4)  from  (3),  we  have 

€•>  b.j  —  c^b  X 
hence  ^=^ 4^'  (5) 

Now  multiply  (1)  hy  a^  and  (2)  by  «i,  whence 

a^ao^x-\-cCob^y=a^c^  (6) 

a^a^x+a^b<^y=^a^c^  (7) 

Subtracting  (6)  from  (7),  we  have 

hence  ^^^iflZI^i.  (3) 

It  is  to  be  noticed  that  the  denominators  in  the  values 
of  X  and  y  (equations  (5)  and  (8))  are  just  alike  and 
contain  only  the  coefficients  of  x  and  y  in  the  given 
equations  (1)  and  (2).     This  expression  a^b^—ac^b^  is 

called  the  Determinant  of  the  four  quantities^!,  ^2  »^i»  ^2- 
We  agree  to  write  this  determinant  in  the  form  of  a  square, 
where  the  numbers  a^,  b^y  a^,  b^^  are  arranged  in  the 
same  order  as  they  appear  in  the  equations  (1)  and  (2) 
with  a  vertical  line  before  and  after  the  square,  thus : 

^2        ^2 


696 


UNIVERSITY   ALGEBRA. 


In  order  that  this  shall  mean  the  same  as  a^h<^—a<^b^  it 
is  evident  that  we  must  take  the  product  of  the  diagonal 
terms  running  downward  to  the  right,  and  from  this 
product  subtract  the  product  of  the  diagonal  terms  run- 
ning upward  to  the  right. 

Returning  now  to  the  value  of  ;t:  in  equation  (5),  it  is 
seen  that  the  numerator  c^b^—c<^b^  is  like  the  denom- 
inator except  that  ^j  and  c^  are  written  in  place  of  a.^ 
and  ^2*  ^"^  ^^^  expression  is  called  the  determinant  of 
the  four  quantities  ^1,  ^1,  ^2>  ^2>  ^^^>  ^^^^  ^^  other,  may 
be  written 

^1     *i  I 
^2     ^2  I 

provided  it  is  agreed  to  interpret  this  square  array  of 
numbers  the  same  as  the  previous  square  array  was  in- 
terpreted. In  the  value  of  y  in  equation  (8)  it  is  seen 
that  the  numerator  a^Ci—a^c^  is  like  the  denominator 
except  that  c^  and  c.^  take  the  place  of  b^  and  b.^,  and, 
just  as  before,  the  expression  ^^^2— ^2^1  ^^Y  be  written 
in  the  form  of  a  square,  provided  the  square  array  oi 
numbers  be  interpreted  as  before. 

The  values  of  x  and  y,  written  in  the  notation  just 
explained,  are 


x= 


C\ 

^ 

^^2 

b^ 

a, 

b. 

«2 

b. 

and  y= 


a, 

f, 

«2 

<^2 

«1 
«2 

b, 
b. 

Each  of  these  determinants  in  the  numerators  and 
denominators  of  the  values  oi  x  and  y  is  expanded  (J.  e,^ 
written  out  in  the  ordinary  form)  by  the  same  rule,  viz. : 
Multiply  together  the  numbers  in  the  diagonal  rmining 
dow7iward  to  the  right  and  from  this  product  subtract  the 
product  of  the  numbers  in  the  diagonal  rummig  upward  to 
the  right. 


DETERMIMANTS. 


697 


The  student  should  carefully  notice  these  values  of  x 
and  y.  In  each  case  the  denominator  is  the  determinant 
obtained  by  writing  the  coefficients  of  x  and  y  in  the  same 
order  as  they  appear  in  the  given  equatio7is,  while  in  each 
case  the  numerator  is  obtained  from  the  denominator  by 
wr  ti  g  the  right  hand  members  in  place  of  the  coefficients 
of  the  quantity  whose  value  is  sought.  The  results  in  this 
form  are  easy  to  remember  and  convenient  to  use. 


EXAMPLES. 


Find  the  values  of  x  and  y  in  the  following  equations, 
using  the  notation  just  explained: 


I. 


\2x-\-Zy=l2 


Here  jr=  - 


7     -1  1 
12        8 


_21^(-12)_33 
"  9-(-  2)~11~^ 


3  -1  I 
I    3        3| 

3        7 

I    2  12  I  _    36-14 

^-  \    3  -1     ~  9-(-2) 
I    2        3| 

^-     \2x+by=lb 

r3;ir-7j/=10 
3-     Ibx-Zy^^  2 

852.     Definitions. 


22 


4. 


5. 


r3^-f2y=16 

(7;r+q)/=38 

r3;r+5>/=13 
\lx+'6y^lZ 


In  the  determinant 
'^2 


«2        ^2 

the  four  quantities  a^,  b^,  a^,  b^y  are  called  Elements.* 
The  elements  composing  any  horizoiital  line  are  called 
a  Row,  and  those  composing  any  vertical  line  are  called 
a  Column. 


*  These  quantities  are  called  Constituents  in  Salmon's  Modern  Hififher 
Algebra,  in  Chrysta.'s  Algebra,  in  Adlis'  Algebra,  in  Burn^ide  aud  Pautou'a 
Theory  of  Equations,  aud  in  Todhuuter's  Theory  of  Equations. 


698  UNIVERSITY   ALGEBRA. 

When  the  determinant  of  two  rows  and  two  columns  is 
expanded,  i,  e.,  written  out  in  the  form  a^b^—a^b^  each 
term  is  seen  to  contain  two  factors  and  hence  the  deter- 
minant is  said  to  be  of  the  Second  Order.  When  ex- 
panded, each  term  is  seen  to  contain  one  and  only  one 
element  from  each  row  and  one  and  only  one  from  each 
column. 

853.  I/Ct  us  now  find  the  values  of  x,  y,  and  z  from 
three  equations  of  the  first  degree  containing  three  un- 
known quantities. 

a^x+b^y+c^z^d^.  (1) 

a^x-^b.^y+c<2,z=d^.  >        (2) 

a^x-\-b^y-^c^z=d^,  (3) 

Multiplying  (1)  by  b^c^  —  b^c^,  (2)  by  — (^i^s  —  ^s^i) 

and  (3)  by  b^c^  —  b^Ci,  we  obtain 

+  ^1(^2^3— ^3<^2>=^l(^2<^3— ^3*^2)  (4) 

—  ^2(^1^3— ^3^1>— ^2(^1<^3— ^3*^1):^ 

—  ^2(^1^3— ^a^l>=— ^2(^1^3— ^3^1)  (5) 

^3(^1^2— <^2^l)-^+<^3(^1^2—<^2^l)j^ 

4-^3(^1^2— ^2^1>=^3(<^1^2— ^2^1)  (6) 

Adding  (4),  (5)  and  (6),  we  obtain 

[^l(^2'^3—^3<^2)— ^2(^1^3— ^3^1) +'^3(^1^2— <^2^l)]^ 

=  ^l(^2^3--^3^2)-"^2(^J^3— ^3^l)+^3(<^l'^2— ^2^1)     (7) 

^^^1(^2^3— ^3^2)  — ^2(<^1^8"-<^3^l)+^3(^l<^2— ^2^1)       ^ON 
«l(^2^3  — ^3^2)— ^2(^1^3  =  ^8^l)+^3(^1^2~^2^l) 

Similarly  multiplying  (1)  by  -— (<^2^3~"^3^2).  (2)  by 
^1^3 ""^3^1 »  ^^^  (^)  ^y  <^i^2~'^2^i>  ^^^  adding  the  re- 
sulting equations  together,  we  obtain 

—  ^l(^2<^8-^8^2)  +  ^2(^1^3-^3^l)-^8(^1^2-^2^1  )     ^g-x 

—  <^l(^2^3-^3^2)  +  ^2V;^1^3-«3^l)-^8(^i^2-^2^l) 


DETERMINANTS.  699 

Again,  multiplying  (1)  by  <^2^3"~^3*2>  and  (2)  by 
"•(<^i^3""^3^i)j  ^^^  (^)  ^y  <^i^2"~^2^i)  a^^  adding  the 
resulting  equations  together,  we  obtain 

^^l(^2<^3-^3^2)-^2  (^1^3 -^3^1)4-^3  (^1^2-^2<^l)     qqn 
^1(^2^3- ^3^2) -^2(^1^3 -^3^1)  + <^3  (^1^2- «2^l) 

If  we  remove  the  parentheses  from  the  denominators 
in  equations  8,  9,  and  10,  each  of  these  denominators  is 
easily  seen  to  be 

^1<^2^3""^1^3^2+^2^3^1""<^2^1^3+^3^1^2""^3^2^1     (H) 

Now  this  expression  (11)  which  contains  only  the  coeffi- 
cients of  X,  y,  z  in  equations  (1)  (2)  (3),  we  wish  to  write 
in  the  form  of  a  square  array  where  the  letters  appear  in 
the  same  order  as  in  the  three  given  equations,  if  it  is 
possible  to  interpret  such  a  square  array  so  as  to  give  the 
same  result  as  the  one  here  written.  We  must  then,  ii 
possible,  interpret 

a^   ^1  ^1 

^2  ^2  ^2 


*'3    ^3    *'3 


(12) 


so  as  to  give  the  same  result  as  (11). 

The  quantities  composing  the  square  array  will  be  called 
Elements,  any  horizontal  line  will  be  called  a  Row,  and 
any  vertical  line  a  Column,  and  the  square  array  itself, 
or  its  equal  (11),  will  be  called  a  Determinant.  As  each 
term  of  (11)  is  the  product  of  three  quantities,  the  de- 
terminant is  said  to  be  of  the  Third  Order.  Now  it  is 
easy  to  see  that  the  expression  (11)  can  be  obtained  from 
(12)  by  taking  the  algebraic  sum  of  all  the  products  ob- 
tained by  taking  one  and  only  one,  element  from  each  row, 
and  one,  and  only  one,  from  each  column ;  using  the 
sign  +  or  —  before  each  product,  according  as  the  order 
of  subscripts  in  that  product  is  obtained  from  the  natural 


700 


UNIVERSITY   ALGEBRA. 


order  1,  2,  3,  by  an  even  or  an  odd  number  of  interchanges 
of  consecutive  subscripts.     Hence  we  may  write 

a^  di  Ci 

ab^c^-aj)c^^aj)^c^-ab^c^jraj)^c-aj)^c^    (13) 


Each  of  the  denominators  in  (8),  (9)  and  (10)  is  equal 
to  the  determinant  (12);  and,  since  in  (8)  the  numerator 
is  seen  to  differ  from  the  denominator  only  in  having  a^^ 
a^,  a^  replaced  by  d^,  d.^^,  d^,  therefore  the  numerator  in 


(8)  is  equal  to  the  determinant 


a. 

l>i   C\ 

d^ 

b,  c. 

d^ 

b^  Cz 

As  the  numerator  in  (9)  is  seen  to  differ  from  the  denom- 
inator only  in  having  b^,  b^,  bo.  replaced  by  d^,  d^,  d^, 
therefore  this  numerator  equals  the  determinant 

d'l  ^2 


3     ^3 


And  similarly  the  numerator  in  the  value  of  z  in  equa- 
tion (10)  equals  the  determinant 

^1  ^1  ^1 

Cf">     ^2     ^2 


b.  d.. 


Hence  we  have 


x  = 


d. 

b^ 

c^ 

d. 

b. 

Ci 

d. 

^ 

<:% 

a, 

b. 

<^i 

«2 

b. 

Ci 

^3 

bz 

<^3 

a^ 

d. 

Cx 

«2 

d.^ 

Cl 

«8 

d^ 

c.i 

«1 

b. 

c^ 

flj 

b. 

Ci 

«3 

b. 

c» 

«i 

b. 

d. 

«2 

bi 

di 

a, 

b. 

d. 

«1 

b. 

-^i 

«2 

bi 

^2 

«3 

bz 

^3 

It  is  to  be  noticed  that  in  these  values  of  x,  y  and^a*  the 
denominator  in  each  case  is  the  determinant  formed  by 
taking  the  coefficients  of  x,  y  and  z  in  just  the  order  in 


DETERMINANTS.  7OI 

which  they  appear  in  the  three  given  equations,  and  that 
in  each  case  the  numerator  is  obtained  from  the  denom- 
inator by  substituting  therein  the  right-hand  members  of 
the  equations  in  place  of  the  coefficients  of  the  quantity 
whose  value  is  sought. 

EXAMPLES. 

Find  by  determinants  the  values  of  x,  y  and  2  in  the 
following  equations : 

(\x-\-  6j/-3^=17.  r   ;ir+  jK+25'=83. 

1.  \    x+  7j/4-  5'=35.        3.    ]    x+2y+  ^=82. 
(5ji:+13j/4  4^=82.  (2x-\-  y+  ^=31. 

r   x+   2y-hSz=  6.  C   x+Sy-i-  5^=4. 

2.  ]    x-\-   Sy+4z=  8.        4.    ]  2x+5y+   ?,z=2. 
i2x+  5>/+8^=15.  iSx+dy+102=7. 

854.  There  is  no  difficulty  in  extending  the  process 
above  employed  to  a  greater  number  of  equations,  but, 
after  these  illustrations,  we  prefer  to  treat  determinants 
by  themselves,  independent  of  these  application  to  a  set 
of  linear  equations.*  We  will,  however,  return  to  the 
solution  of  a  set  of  linear  equations  when  we  have  obtained 
a  number  of  properties  of  determinants. 

We  have  seen  that  a  determinant  of  the  second  order, 
composed  of  four  elements,  is  written 

^1  ^1 


a^ 


^2 


and  is  defined  as  being  a^b.^-^a^^b  /,  also  that  a  determinant 

of  the  third  order,  composed  of  nine  elements,  is  written 

a^  b^  c 


^2  ^2  ^2 

i  ^3  ^3  ^3 


and  is  defined  as  being  the  expression, 

*  "I,inear  equations"  are  equations  of  first  degree. 


702 


UNIVERSITY  ALGEBRA. 


Similarly,  a  determinaut  ot  the  wth  order  is  composed  of 
n^  quantities,  called  elements,  and  naturally  written  thus : 


«2 

^i 

<^2 

d,. 

■h 

«3 

^3 

c» 

dl- 

■It 

«4 

^ 

Ci 

d,. 

■h 

The  elements  in  a  horizontal  line  are  called  a  row  and 
those  in  a  vertical  line,  a  column. 

The  determinant  is  defined  as  being  equal  to  the  alge- 
braic sum  of  all  products  that  can  be  formed  by  taking 
one  and  only  one  element  from  each  row  and  one  and 
only  one  from  each  column,  the  sign  +  or  —  being  writ- 
ten before  each  product  or  term  according  as  the  order  of 
the  subscripts  in  that  term  is  derived  from  the  natural 
order  by  an  even  or  an  odd  number  of  interchanges  of 
successive  subscripts,  it  being  understood  that  the  letters 
preserve  the  natural  order. 

The  collection  of  terms  written  out  according  to  this 
definition  is  called  the  Expansion  of  the  determinant. 

855.  It  is  to  be  noticed  that  the  elements  are  here 
represented  by  letters  with  various  subscripts.  Obviously 
other  symbols  might  have  been  chosen  to  represent  the 
elements,  but  the  advantage  of  this  notation  consists  in 
the  fact  that  the  position  of  each  element  is  indicated,  the 
letter  showing  the  column  and  the  subscript  the  row  to 
which  any  given  element  belongs.  Take  for  example  the 
element  d^ ;  since  d  is  the  fourth  letter,  the  element 
helongs  in  the  fourth  column,  and  the  subscript  being  3, 
it  belongs  to  the  third  row;  thus  its  position  in  the  deter- 
minant is  completely  indicated.  Of  course,  the  position 
of  an  element  would  not  be  thus  indicated  if  there  should 


DETERMINANTS.  703 

be  any  disarrangement  in  the  rows  or  columns  of  the 
above  determinant,  unless  we  knew  exactly  what  disar- 
rangement had  taken  place. 

856.  Since  from  the  above  definition  each  term  must 
contain  one  and  only  one  element  from  the  first  column, 
therefore  each  term  must  contain  a  with  some  subscript; 
and  since  each  term  must  contain  one  and  only  one  element 
from  the  second  column,  therefore  each  term  must  contain 
b  with  some  subscript.  In  the  same  way  it  follows  that 
each  term  must  contain  ^  with  some  subscript,  and  so  for  the 
other  letters.  Hence  all  letters  appear  in  each  term  of  th^ 
expansion  and  no  letter  appears  more  than  once  in  the 
same  term. 

Again,  from  the  definition,  each  term  in  th^  expansion 
must  contain  one  and  only  one  element  from  the  first 
row;  therefore  each  term  must  contain  some  letter  with  a 
subscript  1 ;  and  since  each  term  contains  on^  element 
from  the  second  row,  therefore  each  tei^i  miisK:  contain 
some  letter  with  a  subscript  2;  and  in  the  «ame  way  each 
term  must  contain  some  letter  with  the  subscript  3,  and 
so  for  the  other  subscripts.  Hence  all  the  subscripts  ap- 
pear in  each  term  of  the  expansion  and  no  subscript 
appears  more  than  once  in  the  same  term. 

From  this  it  is  seen  that  every  term  in  the  expansion 
of  the  determinant  contains  all  the  letters  a,  b,  c,  ■  .1 
with  all  the  subscripts  1.  2,  8,  •  •  •;?,  and  if  we  please  we 
may  keep  the  letters  in  their  natural  order  and  the  sub- 
scripts will  be  attached  to  these  letters  in  eveTy  possible 
orderi     To  illustrate >  a  determinant  of  the  fourth  order 


^1  ^1  ^1  ^\ 

^2    ^2    ^2  ^2 

^3    ^3    ^3  ^3 

a^  b^  c^  d^ 


704  UNIVERSITY    ALGEBRA. 

expanded  becomes 

—ab,i\d^  +  ^//;^3  +  ^2  Vx^4  —  ^.  ^4^  ~"  ^«^4^i^3  +  ^2  Vs^x 
+  ^3^x^=<,  -  ^3^z^4<  -  ^3  Vt^4  +  ^3^4^  ^  ^3  Vx^ ^ ^3  V/x 

In  this  expansion  the  signs  are  determined  according 
to  the  definition  by  the  order  of  subscripts.  Take  for  ex- 
ample the  term  a^b^c^d^,  where  the  subscripts  appear  in 
the  order  42ol.  This  order  can  be  determined  from  the 
natural  order,  as  follows: 

Natural  order  12  3  4 

First,  interchange  4  and  3  12  4  3 
Second,  interchange  4  and  2  14  2  3 
Third,  interchange  4  and  1  4  12  3 
Fourth,  interchange  2  and  1  4  2  13 
Fifth,  interchange  3  and  1  4  2  3  1 
As  5  is  an  odd  number,  the  sign  before  a^b^c^d^  must 
be  -. 

857.  The  rule  for  determining  the  sign  of  any  term 
in  the  expansion  of  a  determinant  may  be  simplified  by 
noting  that  the  interchange  of  any  two  numbers,  how- 
ever far  removed,  is  the  same  as  an  odd  number  of  inter- 
changes of  successive  numbers.    For  suppose  any  number 

of  numbers, 

12  34.../^    >  >m-  >  ., 

Let  there  be  r  numbers  between  k  and  m.  Then,  if  we 
wish  to  interchange  fit  and  h,  we  have  to  pass  m  to  the 
left  successively  over  the  r  intervening  numbers,  and  then 
over  the  h ;  we  have  next  to  pass  h  to  the  right  over  the 
r  numbers  that  originally  separated  h  and  77i.  In  passing 
m  to  the  left  we  have  made  r-f  1  interchanges  of  succes- 
sive numbers,  and  in  passing  h  to  the  right  we  have  made 
r  interchanges,  so  in  all  there  are  2r+l  interchanges  of 


DETERMINANTS. 


705 


successive  numbers,  and  this  is  an  odd  number  whatever 
be  the  value  of  r.  We  may  then  strike  out  the  word  suc- 
cessive in  the  rule  of  determining  the  sign  of  any  term  in 
the  expansion  of  a  determinant. 

The  order  4231,  derived  by  five  successive  interchanges, 
may  be  derived  by  a  single  interchange  of  4  and  1. 

858.  By  using  subscripted  letters  for  the  elements  of 
a  determinant  it  can  be  expanded  with  considerable  ease, 
but  how  can  the  terms  be  written  out  when  other  symbols 
.are  used  to  denote  the  elements  ?  Suppose  we  wish  the 
expansion  of 

Now  we  know  that  the  expansion  of 
«i   ^\  ^1 


a 

b 

c 

d 

e 

f 

g 

h 

k 

(1) 


''2     ^2    ^2 
«3    ^3    ^3 


(2) 


is    «i^2^3""^1^3^2  — ^2^1^3  +  ^2<^3^1    +  ^3^1^2  —  ^3^2^1 , 

and  to  obtain  the  expansion  of  (1)  we  must,  of  course, 
substitute  for  each  of  the  elements  in  (2)  that  one  which 
in  (1)  occupies  the  same  position.  The  expansion  of  (1) 
then  becomes 

aek—ahf—dbk + dhc-^-gbf—gec, 

A  better  method  will  be  given  further  on. 

859.     Theorem  I.     77ie  expansion  of  a  determinant 
of  the  nth  order  contains  \n  terms. 

From  the  definition  the  terms  are  obtained  by  taking 
the  letters  a,  b,  c,-  -  -  in  their  natural  order  and  their 
subscripts  in  every  possible  order.  Whence,  the  number 
of  the  terms  is  the  same  as  the  number  of  ways  of  arrang- 
ing: ^  thinsrs  taking  all  at  a  time,  which  is  1  •  2  •  3  •  .  «. 
45— u.  A. 


706  UNIVERSITY   ALGEBRA. 

860.  Theorem  II.  If  in  ajiy  determinant  the  rows 
are  changed  iiito  columiis  aiid  vice  versa^  the  value  of  the 
determinant  is  not  changed. 

Let  us  take  the  case  of  a  determinant  of  the  fifth  order. 
We  are  to  prove 


d. 

^1 

b^ 

«3 

bz 

«4 

b. 

«5 

b. 

d, 
d^ 

^4 

— 

d. 

d^ 

^3 

dz 

d. 

<^5 

d. 

h 

<^5 

d. 

e^ 

ei 

e-i 

<?3 

^4 

<?5 

For  convenience,  let  us  represent  the  first  determinant" 
by  A  and  the  second  by  A'.  Now  it  is  evident  that  A  in- 
volves the  subscripts  in  the  same  way  as  A'  involves  the 
letters  and  vice  versa.  A'  equals  the  sum  of  all  products 
with  proper  signs  attached  which  can  be  formed  by  taking 
one  and  only  one  element  from  each  row  and  one  and 
only  one  from  each  column.  Therefore  the  terms  of  A' 
are  numerically  equal  to  those  of  A,  and  if  we  can  show  that 
the  numerically  equal  terms  in  the  two  determinants  have 
the  same  sign  the  theorem  is  proved.  Now  a^b^c^d^e^, 
or  what  is  the  same  thing,  c^d^b^e^a^  is  a  term  of  both 
determinants.  The  sign  of  a-^b^c^d^e^  considered  is  a 
term  of  A  is  determined  by  the  number  of  interchanges 
of  subscripts  required  to  pass  from  the  natural  order  12345 
to  the  given  order  53124  or,  what  is  the  same  thing,  the 
number  of  interchanges  required  to  pass  from  the  given 
order  53124  back  to  the  natural  order  12345,  while  the 
sign  of  c^d^b^e^a^  considered  as  a  term  of  A'  as  deter- 
mined by  the  number  of  interchanges  of  letters  required 
to  pass  from  the  natural  order  abcde  to  the  given  order 
cdbea  or,  what  is  evidently  the  same  thing,  the  number  of 
interchanges  required  to  pass  from  the  given  order  cdbea 
back  to  the  natural  order  abcde. 

We  have  then  to  prove  that  the  number  of  interchanges 


DETERMINANTS.  ^0^ 

of  subscripts  in  one  case  is  the  same  as  tlie  number  of 
interchanges  of  letters  in  the  other  case. 

The  two  expressions  a^b^c^d^e^  and  c^d^b^e^a^  are 
exactly  alike  except  the  order  of  the  factors  in  one  is 
different  from  that  in  the  other.  If  we  interchange  any 
two  subscripts  in  the  first  and  the  letters  to  which  these 
same  subscripts  are  attached  in  the  second,  the  two  re- 
sulting expressions  will  be  alike  except  in  the  order  of 
the  factors.  For  example :  if  we  interchange  the  sub- 
scripts 1  and  5  in  the  first  and  the  letters  a  and  c  in  the 
second,  we  obtain  the  two  expressions 

which  are  alike  except  the  order  of  factors.  Evidently 
the  same  kind  of  operation  maybe  repeated  upon  the  two 
expressions  here  obtained,  and  then  upon  the  two  results 
again,  and  so  on;  and  every  time  we  make  ojze  interchange 
of  subscripts  in  the  first  we  make  one  interchange  oi  letters 
in  the  second.  But  when  we  have  returned  to  the  natural 
order  of  subscripts  in  the  first  we  have  returned  to  the 
natural  order  of  letters  in  the  second,  for  at  each  step  the 
factors  of  one  product  are  the  same  as  those  of  the  other, 
the  only  difference  being  in  the  order  of  those  factors. 
Therefore,  it  requires  the  same  number  of  interchanges  of 
subscripts  in  one  case  as  of  letters  in  the  other.  There- 
fore, the  sign  oi  c^d<2^b^e^a^  considered  as  a  term  of  A'  is 
the  same  as  the  sign  oi  a-^b<^c^d^j^  considered  as  a  term 
of  A. 

The  argument  given  for  the  particular  term  selected 
will  evidently  hold  for  any  term.  Therefore,  the  deter- 
minants A  and  A'  are  equal. 

861.  From  this  it  follows  that  any  theorem  that  can 
be  stated  about  the  rows  of  a  determinant  can  be  stated 
about  the  columns,  and  vice  versa;  for  we  may  write  side 


7o8 


UNIVERSITY    ALGEBRA. 


by  side  two  determinants  in  which  the  rows  of  one  ^re 
the  columns  of  the  other,  and  any  theorem  about  the  rows 
of  one  is  a  theorem  about  the  columns  of  the  other. 

862.  Theorem  III.  If  two  rows  or  two  columiis  of  a 
determviant  are  interchanged  the  sign  of  the  deter^nitiayit  is 
changed. 

Consider  the  two  determinants 
^1     ^1     d_\ 

and 


h 


"-1 
^3 


^2 

^3 
d. 


d, 

d, 
^3 

d. 


Represent  the  first  by  A  and  the  second  by  A',  A'  being^ 
derived  from  A  by  interchanging  the  second  and  fourth 
rows.  Now  take  any  term  of  A'  as  a^^  b^  c^  d^.  This  is 
found  in  A  and  as  in  the  expansion  of  A  the  term  a.^  b^ 
c^  d^  has  the  sign  of  + ,  so  in  the  expansion  of  A'  the 
term  a^  b^  c^  d^  has  the  sign  of  +,  but  a^  b^  c^  d^  is  also 
a  term  of  A  and  as  a  term  of  A  it  has  the  sign  — .  In  the 
same  way  every  term  in  the  expansion  of  A'  appears  in  the 
expansion  of  A  with  an  opposite  sign;  therefore  A=— A'. 

General  Case.  Represent  the  determinant  of  the  wth 
order  by  A  and  from  A  form  another  determinant  by  in- 
terchanging the  k\.\i  and  rth  rows  and  represent  the  new 
determinant  by  A'.  Now  select  ajiy  term  in  the  expan- 
sion of  A',  differing  from  the  selected  term  of  A  only  in 
having  the  >^th  and  rth  subscripts  interchanged,  the  sign 
being  the  same  in  each  case.  This  term  in  the  expansion 
of  A'  also  appears  in  the  expansion  of  A  but  with  an  op- 
posite sign,  being  determined  from  the  previously  selected 
term  of  the  expansion  of  A  by  interchanging  two  sub- 
scripts, which,  of  course,  changes  the  sign.  In  the  same 
way  every  term  in  the  expansion  of  A'  is  found  in  the  ex- 
pansion of  A  with  an  opposite  sign.     Therefore,  A= — A'. 


DETERMINANTS.  7^9 

863.  Corollary.  If  a  determinant  has  tzvo  rows  or 
two  columns  ideiitical^  the  determinant  equals  zero.  For  if 
we  interchange  the  two  identical  rows  or  columns  in  the 
determinant  represented  by  A,  we  get  a  determinant  rep- 
resented by  —A;  but  interchanging  two  identical  rows 
or  columns  cannot  change  the  determinant  at  all.  There- 
fore, A=— A  .-.  2A=0  .-.  A=0. 

864.  Minor  Determinants.  **If  in  any  determinant 
we  erase  any  number  of  rows  and  the  same  number  of 
columns,  the  determinant  formed  with  the  remaining 
rows  and  columns  is  called  a  Minor  of  the  given  deter- 
minant. The  minors  formed  by  erasing  one  row  and 
one  column  are  called  First  Minors ;  those  formed  by 
erasing  two  rows  and  two  columns  are  called  Second 
Minors,  and  so  on.** — Salmon's  Modern  Higher  Alge- 
bra, 

If  the  given  determinant  is  of  the  n\\i  order,  the  first 
minors  are  of  the  («— l)st  order,  the  second  minors  are 
of  the  («  — 2)d  order,  and  so  on.  Thus,  we  may  speak  of 
rth  minors  or  minors  of  the  order  n^-r  indifferently. 
Minors  of  the  first  order  are  the  elements  themselves. 

The  elements  at  the  intersection  of  the  rows  and  col- 
umns erased  also  form  a  minor  of  the  given  determinant 
called  the  Complement  of  the  minor  which  is  left.  In 
any  determinant  the  complement  of  an  rth  minor  is  a 
minor  of  the  rth  order.  If  the  determinant  is  of  the  ^^th 
order  the  complement  of  a  minor  of  the  order  r  is  a  minor 
of  the  order  n—r\  that  is,  the  complement  of  an  rth 
minor  is  an  (;^— r)th  minor. 

In  any  determinant  there  are  as  many  first  minors  as 
there  are  elements  and  the  first  minors  obtained  by  eras- 
ing the  row  and  column  intersecting  in  any  given  element 


7IO 


UNIVERSITY   ALGEBRA. 


is  called  the  first  minor  corresponding  to  that  elemerit.     In 
the  determinant 

cf"i     b<^     C2 

^3     ^3     ^3 

which  we  represent  by  A,  if  we  erase  the  first  row  and 


h  ^3 


first  column,  there  remains   the   determinant 

which  is  the  first  minor  of  A  corresponding  to  a\,  and 

be. 
^     ^     is  the  first  minor  of  A  corresponding 


'1     «-! 
^3    ^6 


similarly 
to  a^'y  also 


to  b^)  and  so  on. 


is  the  first  minor  of  A  corresponding 


865.     Expression  of  a  Determinant  in  Terms  of 
the  Elements  in  any  Row  or  Column. 
Let  the  determinant 


«1 

^ 

<^i 

d. 

^2 

b. 

c^ 

d. 

«3 

h 

H 

dz 

«4 

b. 

ct 

d. 

be  represented  by  A.  Since,  by  the  definition  of  a  deter- 
minant every  term  in  the  expansion  must  contain  some 
element  from  the  first  column,  a  certain  number  of  terms 
in  the  expansion  will  contain  a^,  while  other  terms  will 
contain  a^,  and  so  on.  Collect  together  all  those  terms 
of  A  which  contain  a^ ;  then,  after  taking  out  this  common 
factor  (2  J,  there  will  remain  an  aggregate  of  terms  which 
we  will  represent  by  A^,  so  that  a^A^  will  represent 
the  algebraic  sum  of  all  those  terms  which  contain  a^. 
Referring  to  Art.  856,  we  see  that 

A^  —  b^c^d^  —b^e^d^  — ^3^2^4  +  b^e^d^  +  b^c^d^  —  b^e^d^ . 
In  the  same  manner  we  might  collect  together  those 
terms  in  the  expansion  of  A  which  contain  the  element 


DETERMINANTS.  7II 

^2,  and  represent  the  algebraic  sum  of  all  these  terms 
by  ^2-^2^  then,  from  Art.  856, 

Similarly  the  algebraic  sum  of  all  the  terms  in  the  ex- 
pansion of  A  which  contain  a^  would  be  ^3:3^3  and  the 
algebraic  sum  of  those  terms  containing  a^  would  be  a^A^. 
Now  every  term  in  the  expansion  of  A  must  contain 
some  element  from  the  first  column.  Therefore,  if  we 
collect  into  one  group  those  terms  which  contain  ^1,  and 
into  another  group  those  which  contain  a 2,  and  into  an- 
other those  which  contain  ^3,  and  into  another  those 
which  contain  ^4,  then  in  these  four  groups  we  will  be 
sure  to  have  a/l  the  terms  of  A. 

Therefore,     A=«i^i  +^2^2  +^3^3  +^4^4- 
This  is  an  expression  for  A  in  terms  of  elements  of  the 
first  column.     In  a  similar  way  we  could  obtain  an  ex- 
pression for  A  in  terms  of  the  elements  of  the  second 
column  or  any  other  column  or  any  row. 

If  we  select  the  terms  in  the  expansion  of  A  which 
contain  any  one  of  the  sixteen  elements  and,  after  taking 
out  this  common  factor,  represent  the  remaining  aggre- 
gate of  terms  by  a  capital  letter  of  the  same  name  and 
with  the  same  subscript  as  the  element  we  are  consider- 
ing; then  A  may  be  expressed  in  any  one  of  the  following 
eight  ways : 

A=^l^l-f«2-^2+^3^3+^4^4-  (1) 

A=l^l^,-f^2^2+^3^3+^4^4.  (2) 

A=  C^  Ci  +  ^^2  ^2  +  ^3  Q  +  ^4  Q-  (3) 

A:=d,D,+d2n2  +  d^D^+d^D^.  (4) 

A=ai^i  +  ^ii5i+^iCi+^iZ>i.  (5) 

A=a2^2  +  ^2^2  +  ^2  ^2  +  ^2^2-  (6) 

A=^3^3  4-  ^3^3  +  ^3^  +  ^3  A'  (7) 

A=a^A^+  d,  B^  +  ^4  C4  +  d^D^.  (8) 


712  UNIVERSITY  ALGEBRA, 

The  explanation  is  here  given  for  a  determinant  of  the 
fourth  order,  but  it  is  so  evident  that  the  process  applies 
to  a  determinant  of  any  order  that  we  omit  a  separate  ex- 
planation for  the  general  case. 

866.  Expression  of  a  Determinant  in  Terms  of 
the  First  Minors  Corresponding  to  any  Row  or 
Column. 

As  we  usually  represent  a  determinant  by  A,  so  let  us 
represent  the  first  minor  corresponding  to  ^^  by  A^^  and 
similarly  represent  the  first  minor  corresponding  to  any 
element  by  A  with  that  element  used  as  a  subscript. 

We  will  prove  that  in  the  above  eight  equations  the 
factor  that  multiplies  any  element  is  either  +  or  —  the 
first  minor  corresponding  to  that  element. 
First,  to  prove  ^i=A^^. 

The  terms  of  A  ^  are  obtained  from  the  terms  of  A  that 
contain  the  element  a^  by  striking  out  this  element;  that 
is,  they  consist  of  the  letters  b,  c,  d  in  the  natural  order 
with  the  subscripts  2,  3,  4  attached  to  the  letters  in  every 
order.  Moreover,  in  those  terms  of  A  from  which  the 
terms  of  A  j  are  derived,  the  element  ^j  stands  at  the  head, 
and  hence  the  sign  is  determined  by  the  number  of  inter- 
changes of  the  last  three  subscripts.  But  the  terms  of 
Aflj  also  consist  of  the  three  letters  b,  c,  d  with  the  sub- 
scripts 2,  3,  4  attached  to  these  in  every  possible  order 
and  the  sign  of  each  term  is  here  also  determined  by  the 
number  of  interchanges  of  the  subscripts  2,  3,  4;  hence, 
^,  =  A,,. 

Second,  to  prove  ^2  =  "~^«,- 

The  terms  of  ^2  ^^^  determined  from  the  terms  of  A 
that  contain  the  element  a 2  by  striking  out  this  element; 
that  is,  they  consist  of  the  letters  b^  c,  d  with  the  sub- 
scripts 1,  3,  4  attached  to  these  letters  in  every  possible 


DETERMINANTS.  7^3 

order.  The  sign  of  any  of  these  terms  is  the  same  as  the 
sign  of  the  corresponding  term  in  the  expansion  of  A  and 
in  this  term  of  A  it  requires  one  interchange  of  subscripts 
to  begin  with  to  get  the  subscript  2  or  the  element  a.^  at 
the  head,  and  so  the  sign  of  any  term  oi  A.^\s  determined 
by  a  number*  one  greater  than  the  number  of  subsequent 
interchanges  in  the  subscripts  1,  3,  4. 

The  terms  of  Aaj  also  consist  of  the  letters  5,  r,  d  with 
the  subscripts  1,  3,  4  attached  to  the  letters  in  every  pos- 
sible order;  but  the  sign  of  any  term  of  Aag  is  determined 
by  the  number  of  interchanges  of  subscripts  1,  B,  4; 
hence,  the  sign  of  any  term  in  Aa^  is  opposite  to  the  sign  of 

a  corresponding  termini  2  J  ^^^  consequently -^2""= ^«,- 
In  exactly  the  same  way  it  may  be  shown  that 

^3=  +  A^,  ^,  =  -A,,,         ^3  =  -^v 

Hence  we  see  that  the  multiplier  of  any  element  in  the 
expansion  of  a  determinant  is  either  +  or  — ,  the  first 
minor  corresponding  to  that  element,  the  sign  +  or  — 
being  used  according  as  the  element  is  removed  an  even 
or  odd  number  of  steps  from  the  element  in  the  upper  left 
hand  corner;  where,  in  counting  the  steps,  we  pass  along 
the  first  row  to  the  right  until  we  are  in  the  column  in 
which  the  element  is  found,  and  then  downward  until  we 
come  to  the  element,  but  never  pass  along  a  diagonal  line. 

Calling  that  diagonal  running  from  the  upper  left  hand 
corner  to  the  lower  right  hand  corner  the  Leading  or 
Principal  Diagonal,  then  the  rule  just  given  to  determine 
the  sign  may  be  simplified.  The  sign  +  or  --  is  used 
according  aa  the  element  taken  is  an  even  or  odd  number 
of  steps  from  any  element  in  the  principal  diagonal. 

•The  8ig3i  si  j^us  is  this.number  is  even  and  minus  if  this  number  is  odd. 


7H 


UNIVERSITY   ALGEBRA. 


All  this  is  given  for  a  determinant  of  the  fourth  order; 
but  a  careful  examination  of  the  discussion  will  show  that 
it  is  equally  applicable  to  a  determinant  of  the  «th  order. 

867.  The  above  gives  us  a  new  way  of  expanding  a 
determinant.  Take,  for  instance,  a  determinant  of  the 
fourth  order  and  express  it  in  terms  of  the  first  minors 
corresponding  to  the  elements  of  any  row  or  column,  say 
the  first  column : 


A= 


Now  we  can  expand  each  of  these  determinants  of  the 
third  order  in  terms  of  their  first  minors  in  the  same  way. 


«3^3^3^3 
«4VA 

=  «x 

v/4 

-«. 

+''3 

Kc,d, 

-«4 

Kc.d, 

W3 

a\yi. 

-K 

c^  d^ 
Cid^ 

+  ^4 

f  2  d^ 
Cz  d^ 

-«2  I  f>X 

c^  d^ 

-h 

c^  d^ 
c^  d^ 

+  *4 

c,  d, 
Cz  d^ 

+«» { ^ 

c-i  d^ 
Ci  d^ 

-b. 

c,d, 
c^d^ 

+^ 

^2  d.^ 

— «4  1  *1 

c,  dj 
^i  d% 

-b. 

+  -^3 

C\  dx 

Ci  d^ 

Expanding  each  of  these  twelve  determinants  of  the 
second  order  we  have  the  expansion  of  the  determinant 
A  as  follows: 

)— ^2V3<  +  ^2V/3  +  ^«Vx<-"^«V4<--^»Vx^3  +  ^.Wx 

+  «3V2^4-SV4<~«3Vx<  +  ^3V/x  +  ^3Vx<-^3Va^x 

(-^V2^3+^V3<  +  ^^M-^4V3<-^4Vx<  +  ^4Va^. 

The  result  agrees  with  the  expansion  given  in  Art.  856. 

By  the  process  here  given  we  can  expand  a  determinant 

when  the  elements  are  represented  by  any  symbols  what- 


A= 


DETERMINANTS. 


715 


ever  as  easily  as  when  the  elements  are  represented  by 
letters  with  subscripts  :  thus 

be  I        /-I 

/I   -d 


=  a 


e 
h 


b 
h 


+  g 


c 
f 


^aiek—hf^-dibk-hc)  -\-g{hf-'ec) 
=  aek^ahf  —  dbk-\-  dhc+gbf—gec. 


EXAMPl^KS. 


I.     Express  the  determinant 


a 

b 

c 

d 

e 

f 

g 

h 

k 

in  terms  of  the  minors  corresponding  to  the  elements  in 
the  second  column  and  expand  the  resulting  determinants 
of  the  second  order  and  show  that  the  result  agrees 
with  that  in  the  last  article. 

2.  Express  the  same  determinant,  in  terms  of  minors 
corresponding  to  the  elements  in  the  second  row  and 
show  that  the  final  result  is  the  same  as  before. 

3.  Find  the  value  of  each  of  the  following  determinants: 


1    2    3 

1    3    1 

12    3! 

13    2 

3    3    3 

2    3    4 

14    4 

14    4 

14    4 

3    3    4 

3    3    3 

3    3    3 

868.  Theorem  IV.  If  all  the  elements  of  any  row  or 
column  can  be  expressed  as  the  sum.  of  two  or  more  quanti- 
ties, the7i  the  determinant  can  be  expressed  as  the  sum  of  twa 
or  more  determinants. 

Take  for  example  the  determinant 

(^i+Z^i)  ^1  ^1 
(^2+^2)  ^2  ^2 


7i6 


UNIVERSITY   ALGEBRA. 


Expressing  this  in  terms  of  minors  corresponding  to  the 
elements  of  the  first  column,  we  get 

b.  c, 


K+)3i) 


(«2-f^2) 


V 


+  -^^1 


^3   ''S 


2    -a. 


by  Cy 


■\b. 


\-?^ 


by  Cy 
bt  c. 


2   ^2    I    I 

1   ^1    I    I 
J^2    I    j 


^ 
^ 


But  the  expression  in  the  first  bracket  is  evidently  equal 
to  the  determinant 

ttj   b^   c^ 

0-2    ^2    ^2 
«3    ^3    ^3 

and  the  expression  in  the  second  bracket  equals  the  de- 
terminant 


Hence 


If  the  given  determinant  had  been 

(a2+/^2+72)  ^2   ^2 
{^Z^Pz  +  yz)   ^3  ^3 

then,  by  what  has  just  been  given,  this  determinant  is 
equal  to 

(ai+/3i)  b^  c^ 

(^2+^2)  ^2  ^2 

(a3+i^3)   ^3  ^3 

and  supplying  the  value  of  the  first  of  these  from  above, 
we  have 


P 

,  ^ 

^1 

Pi  bi  c^ 

Pz  ^3  ^« 

(ai+^,)^^l 

a,  by 

<^i 

iS. 

*1  <"l 

(«2+/32)  *2   Ci 

= 

02  ^2 

<^2 

+ 

/J2 

ba  C2 

(as +^3)   ^8 

<^3 

«3 

^ 

<^8 

^3 

I'zC^ 

71  ^1  ^1 

+ 

72  ^2  ^2 

73  ^3  ^3 

(«i+^i+7i)  ^i  fi 
(^2+^2+72)  <^2  ^: 

(a8+^8+73)  ^8  ^1 


.1 

«1   ^)    ^1 

Pi   ^1   ^1 

7i  ^i  ^ 

"2 

= 

^2    ^2   ^2 

+ 

/^2    ^2   ^2 

+ 

72    ^2  ^^2 

8 

«3    ^8   ^3 

/?3   ^8   ^8 

73   ^8  ^3 

DETERMINANTS. 


717 


Similarly  if  all  the  elements  of  the  first  column  were  the 
sum  of  any  number  of  quantities,  the  determinant  would 
equal  the  sum  of  the  same  number  of  determinants,  the 
forms  of  which  are  evident  from  the  example  here  given. 

Evidently  this  peculiarity  might  have  presented  itself 
in  any  other  column  as  well  as  the  first,  or  in  any  row. 

A  precisely  similar  discussion  would  show  that  if  all 
the  elements  of  any  row  or  column  were  expressed  as  the 
difference  between  two  quantities,  then  the  determinant 
could  be  expressed  as  the  diiBference  between  two  deter- 
minants. 

KXAMPI,^. 

Express  the  determinant 

2  3  1 

3  3  3 

4  4  1 

as  the  sum  of  two  determinants  in  three  different  ways ; 
find  the  value  of  each  of  the  resulting  determinants  and 
compare  the  sum  with  the  value  of  the  given  determinant. 
Also  express  it  as  the  sum  of  three  determinants,  find  the 
value  of  each,  and  add. 

869.  Theroeni  V.  If  all  the  elements  of  any  row  or 
column  be  micltiplied  by  a  common  factor  the  determinant  is 
mnltiplied  by  that  factor. 

Let  us  take  the  following  determinant,  which  we  rep- 
resent by  A: 

a  b  c 
d  e  f 
g  h  k 

and  multiply  all  the  elements  in  the  first  column  bj''  /», 
then  we  obtain 


ma 

b  c 

VI  d 

e  f 

tng 

h  k 

7i8 


UNIVERSITY   ALGEBRA. 


Calling  this  A',  expressing  A  and  A'  in  terms  of  the  minors 
corresponding  to  the  first  column,  we  get 


A=« 

e  f 
hk 

■       b  c 
-•^  hk 

+sr 

b!=zma 

e  f 
h  k 

-'Vk 

+mg 

c 

f 

c  I 

/I 


from  which  it  is  evident  b!^=mt^. 

Corollary  i .  If  all  the  elements  in  any  row  or  column 
contain  a  common  factor,  that  factor  may  be  taken  out  oi 
each  of  the  elements  and  placed  as  a  factor  of  the  remain- 
ing determinant. 

Corollary  2.  Multipljdng  any  row  or  column  by  any 
number  and  dividing  another  row  or  column  by  the  same 
number  does  not  phange  the  value  of  a  determinant. 


KXAMPL^. 

Verify  each  corollary  in  the  determinant: 
12     3 


870.  Theorem  VI.  If  the  elements  of  any  row  or  col- 
umn, each  multiplied  by  the  same  number,  be  added  to  or 
subtracted  from  the  corresponding  elements  of  another  row 
or  column,  the  value  of  the  determinant  is  not  changed. "^ 

*The  wording  ot  the  theorem  should  be  carefully  noted,  for  if  the  elements 
of  any  row  or  column  be  added  to  or  subtracted  from  the  corresponding  elements 
of  another  row  or  column  multiplied  by  the  same  number  the  determinant  u 
changed. 

If  from  the  determinant 

a  b  c 

de  f 

g  hk 

we  make  another  by  adding  the  elements  of  the  second  column  to  m  times  the 
■elements  of  the  first  column  we  get 

I{ina-Vh\  b  c  \ 
{vtd^e  )e  f\ 
{mg-\-h)  h  k   \ 
and  this  is  just  m  times  the  first  one. 


DETERMINANTS. 


719 


a  b  c 

rnh  b  c 

= 

d  e  f 

+ 

me  e  f 

ghk 

mil  h  k 

Let  us  take  tlie  determinant 

'  a  b  c 
d  ef 
ghk 

and  add  to  the  elements  of  the  first  column  the  correspond- 
ing elements  of  the  second  column  each  multiplied  by  m. 

Thus  we  get 

(^a-\-mb^  b  c 
(^d-\-me)  e  f 
ig+m/i)  h  k 

Now,  because  each  element  in  the  first  column  is  the 
sum  of  two  quantities,  therefore 

(<^  +  ;;^^)  b  c 
(d-\-me)  e  f 
(^g-\-mIi)  h  k 

Taking  out  the  factor  m  from  the  elements  of  the  first 
column  of  the  second  determinant  on  the  right  side  of 
the  equation,  we  get 

(^a-\-mb')  b  c 
(  d+me)  e  f 
Ig+m/i)  h  k 

But  the  last  determinant  in  the  equation  has  two  iden- 
tical columns  and  therefore  vanishes,  whence 
(  a  +  mb)  b  c 
(d+me)  ef 
{g+mli)  h  k 

Similarly 
(  a — m,b)  b  c 
(  d—me)  e  f 
i^g—mli)  h  k 

Scholium. — When  dealing  with  a  numerical  determi- 
nant in  which  the  elements  are  large  numbers,  we  may 
combine  rows  with  rows  and  columns  with  columns  ac- 
cording to  this  theorem  so  as  to  reduce  the  elements  to 


a  b  c 

b  b  c 

= 

def 

+m 

e  e  f 

ghk 

hhk 

a  b  c 

= 

def 

ghk 

a  b  c 

b  b  c 

a  b  c 

C= 

def 

—m 

e  e  f 

= 

def 

ghk 

hhk 

ghk 

720 


UNIVERSITY   ALGEBRA. 


Ismaller  numbers  and  thus  obtain  a  determinant  easier 
to  compute. 

871.  Theorem  VII.  If  all  the  elements  but  one  in 
a7iy  row  or  cohcmn  be  zero,  the  detenninant  may  be  reduced 
to  one  of  the  next  lower  order. 

Take,  for  example,  the  determinant 

«i  b^  0    d^ 
an  b.y  0    do 


b.  0 


.         .  ^3 

^4     ^4    ^4    ^4 

If  this  be  expressed  in  terms  of  minors  corresponding 
to  the  elements  in  the  third  column,  it  evidently  equals 


0 


2  ^2    ^2 

3  h   ^3 

4  <^4  ^4 


a^  b^  d^ 

^1  ^1  ^l 

^1  ^j  d^ 

-0 

^3  ^3  ^3 

a^  b^  d^ 

+0 

«2  ^2  ^2 
^4  ^4  ^4 

-^4 

^2   <^2  ^2 
^3  ^3  ^3 

whi 

ch  equals 

-^4 

a^  b^  d^ 

ac^bc^d.^ 

«3  ^3  ^3 

872.  To  Compute  the  Value  of  a  Numerical  De- 
terminant. If  we  have  to  compute  the  value  of  a 
numerical  determinant  we  first  look  to  see  if  all  the  ele- 
ments of  any  row  or  column  contain  a  common  factor  and 
remove  as  many  common  factors  as  possible  in  order  to 
reduce  the  elements  to  smaller  numbers ;  then  v/e  seek, 
by  some  combination  of  rows  with  rows  or  columns  with 
columns,  to  still  further  reduce  the  elements,  especially 
aiming  to  transform  the  determinant  so  that  in  some  row 
or  some  column  all  the  elements  but  one  shall  be  zero, 
when  the  determinant  may  be  reduced  to  one  of  a  lower 
order.  We  then  treat  the  new  determinant  in  a  similar 
way,  and  thus  by  continual  reductions  we  may  find  its 
value  usually  much  more  easily  than  by  expanding. 


DETERMINANTS. 


721 


I^t  US  compute  the  value  of  the  determinant. 


12  6  3 
6  3  1 
9  4  1 

10  3  2 


12  24  6  4 


9 
2 
2 
5 
12 


Take  the  factor  3  from  the  first  row  and  2  from  the  last 


and  we  get 


4  2  13 

6  3  12 

9  4  12 

10  3  2  5 

12  3  2  6 


Subtract  two  times  the  first  column  from  the  second  col- 
umn; also  subtract  the  fourth  column  from  the  third;  and 
subtract  the  fourth  column  from  the  fifth; 


2 

0 

1 

1 

3 

0 

2 

1 

4 

1 

3 

1 

5 

0 

1 

2 

6 

0 

1 

2 

2  112 

3  2   11 

5  12   3 

6  12   4 


Subtract  the  fourth  column  from  the  first: 


-6 


0112 

2   2    11 
2    12   3 

=  -12 

2   12  4 

0  112 
12  11 
112  3 
112   4 


-12 


oil 

=-12 

1   2   1 

112 

Subtract  the  third  row  from  the  fourth: 

0  112 
12  11 
112  3 
0  0  0   1 

Subtract  the  first  column  from  the  second: 
-12 

46  -  U.  A. 


0 

1 

1 

1 

1 

1 

1 

0 

2 

722 


UNIVERSITY   ALGEBRA. 


Subtract  the  first  row  from  the  second : 


-12 


0   1   1 

1   0  0 

=  12 

1   0  2 

=24 


which  is  the  value  of  the  determinant  of  the  fifth  order 
that  we  started  with. 

873.     In  article  865  eight  different  expressions  were 
given  for  the  determinant 


h 
h 
b. 


^Z 


dz 
d^ 


each  in  terms  of  the  elements  of  some  row  or  some  col- 
umn, and  it  was  noticed  that 


A^^^a^- 


^2 


d^ 


b^    c^   d-^ 

Ao  =  —^a=—     ^3    ^3    d.^ 

b^    c^   d^ 

Keeping  carefully  in  mind  the  meaning  thus  given  to 
the  capital  letters  with  various  subscripts  it  is  evident  that 

^1^1 +^2-^2 +^3^3 +^4-^4 


'•b. 


b^  c^  d.. 

^i  ^1  d^ 

^3  ^3    ^3 

-^2 

b^  C^  <3 

^4  ^4  ^4 

b^  c^  d^ 

+  h. 


^1  ^1  ^1  I         I 
b^  C2  ^2  |~~^4 

^4  ^4  ^4   I  I 


^ 

Cl 

d. 

b^ 

Co, 

do 

b. 

C-i 

dl 

and  this  is  evidently  the  expression  of  the  determinant 
b^  by  c^  d^ 
^2  ^2  ^2  ^2 
^3  ^3  ^3  ^3 

^4    ^4   ^4   ^4 

in  terms  of  the  minors  corresponding  to  the  elements  of 
the  first  column.  Now  this  determinant,  having  two 
identical  columns,  equals  zero ;  hence, 

/^l^l +^2^2 +<^3^3 +^4^4  =  0- 


DETERMINANTS. 


723 


In  the  same  way  we  could  obtain  a  relation  connecting 
the  elements  of  any  row  or  column  with  minors  corres- 
ponding to  the  elements  of  some  other  row  or  column. 
There  would  be  in  all  twenty-four  such  relations  given 
by  the  determinant 

<2j  b^  Ci  d^ 

a  2  ^2  C2  ^2 

^3  ^3  ^3  ^3 

^4  ^4  ^4  ^4 

In  the  same  way  if  we  were  given  the  determinant  of  the 
ni\i  order: 

^1  ^1  ^1    '  *  ^1 

^2    ^2    ^2  •    •    -^2 


3    *^3   ^i 
a^  ^4  ^4  • 


^3 


we  could  obtain  %i  different  expressions  for  it,  each  one 
as  multiples  of  the  elements  of  a  row  or  a  column,  and  we 
could  obtain  %i{7i—\)  other  relations  connecting  the  ele- 
ments of  one  row  or  column  with  the  minors  corresponding 
to  the  elements  of  another  row  or  column.  As  samples, 
we  write  two  equations  of  each  kind  and  leave  the  student 
to  write  others. 


<3^2^2+^2-^2  +^2  Q+  •    • 

a^A^  +  b^B^  -f  ^3(^2+  •  • 
KXAMPI^KS. 


a,,B,=0. 

/3^2=0. 


I.     Write  the  six  different  expressions 
1     2     3 


for 


and  verify  each  expression. 


724 


UNIVERSITY   ALGEBRA. 


2.  Write  the  twelve  other  equations  expressing  the 
relation  between  the  elements  of  one  row  or  column  and 
the  minors  corresponding  to  the  elements  of  another  row 
or  column  of  the  determinant  in  example  1. 

3.  Express  the  value  of 
a     b 
d     e 
g    h 


c 

f 

k 


in  six  different  ways. 

4.  Write  the  twelve  other  equations  expressing  the  re- 
lation between  the  elements  of  one  row  or  column  and  the 
minors  corresponding  to  the  elements  of  another  row  or 
column  of  the  determinant  in  example  3. 

874.  Application  to  the  Solution  of  Simultaneous 
Equations  of  the  First  Degree. 

Let  us  take  n  equations  of  the  first  degree  containing 
n  unknown  quantities. 

an^x-\-boy-\-Co,2:+  •  •  •  -\-l^v--m2, 
a^x+b^y-\-c^2+ -  '    +l^v=m^. 

a,,x  +  b,,y  -\-c„z-\-     •  •  +  l,,v  =  m„. 
Here  we  suppose  that  the  determinant  formed  by  the 
coefficients  of  the  unknown  quantities,  viz.: 


a^  0^  Ci  • 
^2  ^2  ^2  ' 
«3  ^3  ^3  • 


•/2 


is  not  zero. 


a„  b„  c„  '  '  •  l„ 

Multiplying  the  first  equation  by  A^,  the  second  by  ^^2, 

and  so  on,  we  have 

a^A^x+b^A^y-\-c^A^2-\'  .  •  .  -\-l^A-^v—77i^A ^ 
a<2,A<j^x-\:b^A^y+c^A^2+  •  .  .  +l^A.,v=m,2A^ 
a^A^x-\-b^A^y+c.^A^2+  •  .  .  +l^A^v=^m^A^ 


a„A„x  +  b„A„ y -}- c,,A,,z  +  -  .  .  +l„A„v=m„A„. 


DETERMINANTS.  725 

Adding,  we  obtain  for  the  coefiBcient  of  x  the  determinant 
of  the  coefficients  of  all  the  unknowns,  which  we  will 
represent  by  A;  the  sum  of  the  coefficients  of  each  of  the 
other  unknowns  becomes  zero  (see  Art.  873);  and  the 
right-hand  member  is  what  A  becomes  when  the  <2*s  are 
replaced  by  the  in's-,  that  is,  by  the  right-hand  members 
of  the  given  equations.    Let  us  represent  this  determinant 

Ai 

by  A^.  Then   A;i;=Aj;  therefore,  ;r=-^- 

In  the  same  way,  if  we  multiply  the  first  equation  by 

B^,  the  second  by  ^2>  ^^^  so  on,  and  add  the  resulting 

^% 
equations,  we  get  -^~  V' 

where  A  2  is  what  A.  becomes  when  the  ^'s  are  replaced 
by  the  right  members  of  the  given  equations. 

Again,  if  we  multiply  the  first  by  Cj,  the  second  by 
^2,  and  so  on,  and  add  the  resulting  equations,  we  get 

^3 

2'==— 2. 

A 
where  A3  is  what  A  becomes  when  the  r*s  are  replaced  by 
the  right  members  of  the  given  equations. 

It  is  now  evident  that  the  value  of  any  unknown  quan- 
tity in  the  given  set  of  equations  is  the  ratio  of  two 
determinants,  in  which  the  denominator  is  the  determi- 
nant of  the  coefficients  of  the  given  equations  and  the 
numerator  is  what  the  denominator  becomes  when  the 
right-hand  members  are  put  in  place  of  the  coefficients  of 
the  quantity  whose  value  is  sought. 

875.     Another  Method. 

Form  the  determinant 

(a2^+^2JV+  •  •  •  +hv—m^     b<^     c^'  '  'I2 
(a„x+d„y+  •  •  •     l„v—m„)     d„     c„-  -  ■  /„ 


726 


UNIVERSITY   ALGEBRA. 


Each  element  in  the  first  column  is  formed  by  trans- 
posing the  right-hand  members  of  the  given  equations, 
and  hence  each  of  these  elements  equals  zero,  therefore 
the  determinant  itself  equals  zero.  Now  each  element  in 
the  first  column  is  expressed  as  the  algebraic  sum  of  ;^  + 1 
quantities,  hence  the  determinant  can  be  expressed  as  the 
sum  of  72  +  1  determinants.     Whence 


ax 


b  c 
I   I 


KCn'   •   •/. 


Ky  K^. 


+...- 


bc^- 


ninbnCn*    •    • /. 


=0. 


b„y  b,,  r« .  . .  4 

All  these  determinants,  except  the  first  and  last,  vanish; 
for  after  taking  out  the  common  factor  from  the  first  column 
we  have  left  a  determinant  with  two  identical  columns. 
Moreover,  after  taking  out  the  common  factor  x  from  the 
first  column  of  the  determinant,  we  have  left  the  deter- 
minant A,  and  the  last  determinant  is  evidently  what  we 
have  called  A^;  hence  A;c— A^=0,  or  A;i:=Ai,  or 

_Ai 

■^^  A 
the  same  as  before.     Similarly,  the  other  unknown  quan- 
tities may  be  found. 

876.  If  the  determinant  called  A  should  equal  zero, 
we  cannot  obtain  a  definite,  finite  set  of  values  of  the  un- 
known quantities.  If  all  the  numerators  are  also  zero, 
the  unknown  quantities  are  i7tdeterminate  diVidi  the  equations 
given  are  not  independent.  If,  however,  some  of  the 
numerators  are  not  zero,  then  the  equatiois  cannot  be 
satisfied  by  any  finite  set  of  values,  in  which  case  the 
equations  are  said  to  be  incompatible. 

Thus,  the  equations 

2;t:+  y+  5^=19 

7jir-f4y+ 12^=51 


DETERMINANTS.  ^2^ 

are  independent  and  compatible  and  therefore  form  a 
solvable  set  of  equations.  In  this  example  A=— 2, 
Ai  =  — 2,  A2  =  — 4,  A3  =  — 6,  whence  :r=l,  ^^=2,  ^r=3. 

The  equations 

2x-\-  y+  5^=19 

Sx+2y+  4^=19 

7;r+4j/4- 14^=57 
are  compatible  but  are  not  independent,  the  third  being  de- 
rived from  the  other  two  by  adding  the  second  to  twice  the 
first.  In  this  example  A=0,  Ai  =  0,  A2=0,  A3=0,  so  that 
the  value  of  each  unknown  quantity  assumes  the  inde- 
terminate form  TT- 

The  equations 

2x+  y+  52'=  19 

Zx-\-2y+  4^=19 

7;c+ 47+ 14^=50 
are  incompatible  with  one  another.    If  we  add  the  second 
to  twice  the  first,  we  get 

7;i:+4j/4-14i?=57, 
but  this  contradicts  the  third  equation.     In  this  example 
A=0,  Ai  =  42,  A2  =  — 49,  A3  =  — 7,  so  that  no  finite  values 
of  X,  y,  z  satisfy  the  equations. 

877.     Let  us  now  take  n  equations  of  the  first  degree 
containing  n—\  unknown  quantities: 

a^x-\-b^y-\-c^z-\-  •  •  •  -f /i=0 

«2-^+^2j^+^2'2'+ •••  4-/2=0 
a^x-\-b^y-\-c^z-\r  •  •  •  +/3=0 


dnX  -f  bny-\-  C„Z+  -   '  '  +  4  =  0 

The  absolute  terms  are  written  on  the  left-hand  side  of 
the  equations  because  it  is  better  for  the  method  here 
pursued.  It  is  to  be  noticed  that  the  number  of  equations 


728 


UNIVERSITY   ALGEBRA. 


is  one  more  than  enough  to  enable  us  to  find  the  values 
of  the  unknown  quantities.  Represent  the  determinant 
of  the  known  quantities  by  A,  whence 


A= 


^1  ^1  ^1  '  '  'h 

^2    ^2   ^2  *   *   *  ^2 
^n    "n    ^n  '    *    '  *' n 


Now,  if  we  add  to  the  last  column  x  times  the  first,  y 
times  the  second,  z  times  the  third,  and  so  on,  the  deter- 
minant is  not  changed  in  value;  therefore 


A= 


K  Cn'  *  '  {a„x-\-b^y-\-  c,,2-\-  .  .  .  -f  4) 
But  from  the  given  equations  it  is  evident  that  every 
element  in  the  last  column  equals  zero.     Therefore, 

A=0. 
We  have  tacitly  assumed  that  the  given  equations  were 
compatible  with  one  another  and  have  shown  that  the 
determinant  equals  zero;  whence  the  following  theorem: 
If  n  equations  of  the  first  degree  containing  7i  —  1  un- 
known quantities  are  compatible  zvith  one  another  the  deter- 
minant of  the  known  numbers  equals  zero.  But  this 
determinant  may  be  zero  when  the  equations  are  incom- 
patible, as  in  the  set 

'1x-\-  y-\-  4^—16=0 

Sx-\-  y+  2^—11=0 

Sx+Sy-h  8<^— 38=0 

7^+3)/+10^-40=0 

2  14  -16 

3  12  -11 
8  3  8  -38 
7  3  10  -40 

Subtracting  twice  the  second  row  from    the   third  we 
have   two   identical   rows;    hence,    A=0.      But   if  two 


Here  A= 


DETERMINANTS. 


729 


times  the  fisrt  equation  be  added  to  the  second,  we  get 
7^+3y+10<3'— 43=0,  which  contradicts  the  fourth  equa- 
tion, whence  the  system  is  incompatible. 

Let  us  see  if  we  can  tell  when  several  equations  are 
compatible.  For  the  sake  of  definiteness  in  language  let 
us  take  four  equations, 

^2-^+^2  J^  +  ^2^  +  ^2=0 
«4^+<^4J^  +  ^4'S'  +  ^4  =  0 

and  suppose  three  of  these,  as  for  instance  the  last  three, 
are  independent  and  compatible.  Then  from  these  three 
we  find  the  values  of  x,  y,  z  to  be 


x=^ 


Changing  numerators  so  that  the  column  of  ^'s  shall  be 
the  last  column  in  each  case  and  taking  out  the  factor 
—  1  if  it  occurs,  we  get 


jr=- 


These  determinants  are  easily  seen  to  be  minors  of 


-^2  b^  ^2 
-dz  ^3  ^z 
—d^  b^  c^ 

^2  -^2  ^2 
^3  ~^8  ^Z 
^4  -^4  ^4 

,  ^  = 

^2  bc^—d^ 

^3  ^3-'^3 
^4   ^4-^4 

«2  ^2  ^2 
^3  <^3  ^3 
^4    ^4   ^4 

y  y  — 

a^  ^2  ^2 
^3  ^3  ^^ 
^4  ^4  ^4 

^2    ^2    ^2 
^3    ^3   ^3 
^4    ^4   ^4 

^2    ^2    dn 

b^  c^  d^ 

^4    ^4    ^4 

,    J^  = 

(^2    ^2    ^2 
^3   ^H   ^3 
^4   ^4   ^4 

,    '2'=  — 

^2    ^2    ^2 
^3    ^3    ^3 

^4  b^  ^4 

^2    ^2    <^2 
«3    ^3   ^3 
^4    ^4    ^4 

«2    ^2    ^2 
^3    <^3   ^3 
^4    /^4    ^4 

^2    ^2     ^2 
^3    ^3     ^3 
^4    ^4     ^4 

Representing  this  by 
previous  notation. 


x=^  — 


b\  ^1  ^1 
2     2    2     ** 

^4    (^4   ^4   c/4 

A,  we  may  write  according  to 


Vx 


J/; 


^<?. 


^<*x 


730  UNIVERSITY  ALGEBRA. 

Now  if  these  values  be  substituted  in  the  first  of  the  given 
equations,  we  get 

or  ^^A^^—^^A^^+r^A^^— ^^A^^=0, 

but  this  is  evidently  A;  therefore  A=0.  Hence,  if  the 
values  of  x,  y,  z  that  satisfy  the  last  three  equations  also 
satisfy  the  first,  then  A=0;  but  if  the  values  of  ;i:,  y,  z 
found  do  not  satisfy  the  first  equation,  then  A  is  not  zero. 
Our  conclusion  may  be  stated  in  general  language,  as 
follows:  If  we  have  n  linear  equations  containing  n—X 
unknown  quantities,  and  if  some  n^\  of  these  equations 
are  independent  and  compatible,  then  the  determinant  of 
the  known  numbers  equated  to  zero  shows  that  the  n 
equations  are  compatible. 

879.     If  we  are  given  four  homogenous*  equations  of 
the  first  degree  containing  four  unknown  quantities,  as 
a^x+b^y-^-c^z+dyW^^^ 


a^x-j-d^y+c^z-j-d^w=0 

X      V     z 

we  may  divide  by  w  and  then  consider  — ,  ^,  —  the  un- 

w    w    w 

known  quantities  and  we  have  the  case  previously  con- 
sidered. Hence,  if  we  have  n  homogeneous  equations  of 
the  first  degree  containing  n  unknown  quantities,  and 
some  n—\  of  these  equations  are  independent  and  com- 
patible, the  determinant  of  the  coefficients  equated  to  zero 
shows  that  the  n  equations  are  compatible;  or  thus  —  if 
the  determinant  of  the  coefficient  equals  zero  the  equa- 
tions are  compatible  provided  some  n^\  of  them  are 
independent  and  compatible. 

Homogeneous   equations  may  always  be  satisfied  by 


•The  word  "homogeneous"  is  here  used  in  its  strict  sence,  meaninRf  an 
equation  whose  terms  are  of  the  same  degfree  and  with  no  absolute  term,  Thns 
ax-i  hy^  is  homogeneous,  but  ax-Vby-\-c=^  is  not  homogeneous. 


DETERMINANTS.  731 

making  each  of  the  unknown  quantities  zero,  but  this 
solution  is  excluded  and  the  determinants  of  the  coeffi- 
cients equal  to  zero  shows  that  the  equations  are  satisfied 
by  values  other  than  zero  provided  some  n — 1  of  the 
equations  are  independent  and  compatible. 

In  the  applications  of  this  theorem  (which  are  numer- 
ous) it  is  assumed  without  statement  that  some  n—\  of 
the  equations  are  independent  and  compatible  and  the 
usual  statement  is  that  if  the  determinant  equals  zero  the 
equations  are  compatible. 

RATIONAI^IZATION  OF  ANY  AI^GKBRAIC  KXPRKSSION. 

880.   In  Art.  361  it  was  shown  that  a  rational  integral 

function  of  any  surd,  say  ap'  may  be  reduced  to  the  form 
p—\  p—1  1 

A^a-J  -^rA^a  /  +  •  •  • +^/_2^>+^/-)  (1) 

We  shall  now  show  that  a  rationalizing  factor  may  be 

found  for  (1). 

The  notation  will  be  simplified  if  we  give  a  particular 

value  to  p.     For  example:  let  ^=4.     It  will  be  easily 

seen   afterwards  that  the  same  method  may  be  applied 

whatever  the  value  of  p. 

I^et  ^0^^+^l^^  +  ^2^"^  +  ^3=^ 

JJ  2  1  1 

then  ^ia^+-^2^^+^3^^+^o  =  ^^^^^> 

also,  ^2^^  +  -^3^^  +  ^0^^  +  ^l=^^^^> 

3  2  1  3 

and  A^a'i-V  A  ^a'^-\-  A  ^a'^  -^^  A<2^^ma'^. 

I  2  3 

Transposing  vi^  ma'^,  ma'^,  and  ma'^  to  the  left  mem- 
bers, and  forming  the  determinant  of  the  left  members, 

we  have,  by  Art.  878, 

Aq  A^  A2  (A^—m) 

Ay  A2  A^  (A^—ma^     ^^ 

A2  A.^  Aq  {Ay—PtaT) 

A^  Aq  Ay  {A^—ma"^ 


732 


UNIVERSITY  ALGEBRA. 


Whence,  b}^  breaking  the  determinant  up  into  the  differ- 
ence of  two  determinants,  we  have 


Aq  A^  A2  1 

Aq  Ai  A2  A^ 

A^  A  2  A^  aJ 

A I  A2^A^  Aq 

A^  A^  Aq  aJ 

^■2  A.^  Aq  A^ 

A^  Aq  A^  aJ 

^3  Aq  A^  A^ 

The  determinant  on  the  right  side  is  rational  as  far  as 
aJ  is  concerned.  Therefore,  the  determinant  on  the  left 
side  is  the  rationalizing  factor  for  niy  as  it  is  the  multiplier 
that  renders  m  rational. 

lyikewise,  a  rationalizing  factor  may  be  found  for  a 
rational  integral  function  of  any  other  surd. 

If  we  have  a  rational  integral  function  of  several  surds, 
we  may  rationalize  the  expression  as  far  as  one  of  those 
surds  is  concerned,  then  rationalize  with  reference  to  one 
of  the  remaining  surds,  and  so  on  until  the  expression  is 
entirely  rational. 

An  important  result  of  this  discussion  is  the  fact  that 
every  algebraic  equation  can  be  rationalized.  Thus,  the 
solution  of  rational  integral  equations  includes  the  solu- 
tion of  all  algebraic  equations  of  whatever  kind. 


14  DAV  i'^j,  uatRowBD 

*-  ^  t  niped  below,  or 

Kenewedbooksaresub^ec^ 


\§o«oT^''"' 


p 


P306O71 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


